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Relativizing the substructural hierarchy. [Partly based on joint work with a) A. Ciabattoni, K. Terui, b) P. Jipsen, c) R. Hor c k.] Nikolaos Galatos University of Denver July 26, 2011 Nikolaos Galatos, TACL11, Marseille, July 2011

  1. Bi-modules Substructiral logics and residuated lattices Outline Residuated lattices x 1 = x = 1 x , ( xy ) z = x ( yz ) ■ Examples Bi-modules x ( y ∨ z ) = xy ∨ xz and ( y ∨ z ) x = yx ∨ zx ■ Formula hierarchy Submodules and nuclei So, if P is a residuated lattice, then ( P, ∨ , · , 1) is a semiring. [In the Lattice frames Residuated frames complete case, a quantale.] GN FL x \ ( y ∧ z ) = ( x \ y ) ∧ ( x \ z ) and ( y ∧ z ) /x = ( y/x ) ∧ ( z/x ) ■ Gentzen frames Compl - CE ( y ∨ z ) \ x = ( y \ x ) ∧ ( z \ x ) and x/ ( y ∨ z ) = ( x/y ) ∧ ( x/z ) ■ Frame applications 1 \ x = x = x/ 1 Equations ■ Simple rules ( yz ) \ x = z \ ( y \ x ) and x/ ( zy ) = ( x/y ) /z ■ FEP Hypersequents x \ ( y/z ) = ( x \ y ) /z ■ Hyper-frames CE for HFL Relativizing to InFL FMP for InFL DFL FEP for IDFL CE for HDFL Relativising Conuclei Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 6 / 28

  2. Bi-modules Substructiral logics and residuated lattices Outline Residuated lattices x 1 = x = 1 x , ( xy ) z = x ( yz ) ■ Examples Bi-modules x ( y ∨ z ) = xy ∨ xz and ( y ∨ z ) x = yx ∨ zx ■ Formula hierarchy Submodules and nuclei So, if P is a residuated lattice, then ( P, ∨ , · , 1) is a semiring. [In the Lattice frames Residuated frames complete case, a quantale.] GN FL x \ ( y ∧ z ) = ( x \ y ) ∧ ( x \ z ) and ( y ∧ z ) /x = ( y/x ) ∧ ( z/x ) ■ Gentzen frames Compl - CE ( y ∨ z ) \ x = ( y \ x ) ∧ ( z \ x ) and x/ ( y ∨ z ) = ( x/y ) ∧ ( x/z ) ■ Frame applications 1 \ x = x = x/ 1 Equations ■ Simple rules ( yz ) \ x = z \ ( y \ x ) and x/ ( zy ) = ( x/y ) /z ■ FEP Hypersequents x \ ( y/z ) = ( x \ y ) /z ■ Hyper-frames CE for HFL So, for N = P , ( P, ∨ , · , 1) acts on both sides on ( N, ∧ ) by \ and / . Relativizing to InFL FMP for InFL Thus, (( N, ∧ ) , \ , / ) becomes a ( P, ∨ , · , 1) -bimodule. DFL FEP for IDFL CE for HDFL Relativising Conuclei Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 6 / 28

  3. Bi-modules Substructiral logics and residuated lattices Outline Residuated lattices x 1 = x = 1 x , ( xy ) z = x ( yz ) ■ Examples Bi-modules x ( y ∨ z ) = xy ∨ xz and ( y ∨ z ) x = yx ∨ zx ■ Formula hierarchy Submodules and nuclei So, if P is a residuated lattice, then ( P, ∨ , · , 1) is a semiring. [In the Lattice frames Residuated frames complete case, a quantale.] GN FL x \ ( y ∧ z ) = ( x \ y ) ∧ ( x \ z ) and ( y ∧ z ) /x = ( y/x ) ∧ ( z/x ) ■ Gentzen frames Compl - CE ( y ∨ z ) \ x = ( y \ x ) ∧ ( z \ x ) and x/ ( y ∨ z ) = ( x/y ) ∧ ( x/z ) ■ Frame applications 1 \ x = x = x/ 1 Equations ■ Simple rules ( yz ) \ x = z \ ( y \ x ) and x/ ( zy ) = ( x/y ) /z ■ FEP Hypersequents x \ ( y/z ) = ( x \ y ) /z ■ Hyper-frames CE for HFL So, for N = P , ( P, ∨ , · , 1) acts on both sides on ( N, ∧ ) by \ and / . Relativizing to InFL FMP for InFL Thus, (( N, ∧ ) , \ , / ) becomes a ( P, ∨ , · , 1) -bimodule. This split DFL matches the notion of polarity . FEP for IDFL CE for HDFL Relativising Conuclei Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 6 / 28

  4. Bi-modules Substructiral logics and residuated lattices Outline Residuated lattices x 1 = x = 1 x , ( xy ) z = x ( yz ) ■ Examples Bi-modules x ( y ∨ z ) = xy ∨ xz and ( y ∨ z ) x = yx ∨ zx ■ Formula hierarchy Submodules and nuclei So, if P is a residuated lattice, then ( P, ∨ , · , 1) is a semiring. [In the Lattice frames Residuated frames complete case, a quantale.] GN FL x \ ( y ∧ z ) = ( x \ y ) ∧ ( x \ z ) and ( y ∧ z ) /x = ( y/x ) ∧ ( z/x ) ■ Gentzen frames Compl - CE ( y ∨ z ) \ x = ( y \ x ) ∧ ( z \ x ) and x/ ( y ∨ z ) = ( x/y ) ∧ ( x/z ) ■ Frame applications 1 \ x = x = x/ 1 Equations ■ Simple rules ( yz ) \ x = z \ ( y \ x ) and x/ ( zy ) = ( x/y ) /z ■ FEP Hypersequents x \ ( y/z ) = ( x \ y ) /z ■ Hyper-frames CE for HFL So, for N = P , ( P, ∨ , · , 1) acts on both sides on ( N, ∧ ) by \ and / . Relativizing to InFL FMP for InFL Thus, (( N, ∧ ) , \ , / ) becomes a ( P, ∨ , · , 1) -bimodule. This split DFL matches the notion of polarity . It also extend to � , � . FEP for IDFL CE for HDFL Relativising Conuclei Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 6 / 28

  5. Bi-modules Substructiral logics and residuated lattices Outline Residuated lattices x 1 = x = 1 x , ( xy ) z = x ( yz ) ■ Examples Bi-modules x ( y ∨ z ) = xy ∨ xz and ( y ∨ z ) x = yx ∨ zx ■ Formula hierarchy Submodules and nuclei So, if P is a residuated lattice, then ( P, ∨ , · , 1) is a semiring. [In the Lattice frames Residuated frames complete case, a quantale.] GN FL x \ ( y ∧ z ) = ( x \ y ) ∧ ( x \ z ) and ( y ∧ z ) /x = ( y/x ) ∧ ( z/x ) ■ Gentzen frames Compl - CE ( y ∨ z ) \ x = ( y \ x ) ∧ ( z \ x ) and x/ ( y ∨ z ) = ( x/y ) ∧ ( x/z ) ■ Frame applications 1 \ x = x = x/ 1 Equations ■ Simple rules ( yz ) \ x = z \ ( y \ x ) and x/ ( zy ) = ( x/y ) /z ■ FEP Hypersequents x \ ( y/z ) = ( x \ y ) /z ■ Hyper-frames CE for HFL So, for N = P , ( P, ∨ , · , 1) acts on both sides on ( N, ∧ ) by \ and / . Relativizing to InFL FMP for InFL Thus, (( N, ∧ ) , \ , / ) becomes a ( P, ∨ , · , 1) -bimodule. This split DFL matches the notion of polarity . It also extend to � , � . FEP for IDFL CE for HDFL The bimodule can be viewed as a two-sorted algebra Relativising Conuclei ( P, ∨ , · , 1 , N, ∧ , \ , / ) . Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 6 / 28

  6. Bi-modules Substructiral logics and residuated lattices Outline Residuated lattices x 1 = x = 1 x , ( xy ) z = x ( yz ) ■ Examples Bi-modules x ( y ∨ z ) = xy ∨ xz and ( y ∨ z ) x = yx ∨ zx ■ Formula hierarchy Submodules and nuclei So, if P is a residuated lattice, then ( P, ∨ , · , 1) is a semiring. [In the Lattice frames Residuated frames complete case, a quantale.] GN FL x \ ( y ∧ z ) = ( x \ y ) ∧ ( x \ z ) and ( y ∧ z ) /x = ( y/x ) ∧ ( z/x ) ■ Gentzen frames Compl - CE ( y ∨ z ) \ x = ( y \ x ) ∧ ( z \ x ) and x/ ( y ∨ z ) = ( x/y ) ∧ ( x/z ) ■ Frame applications 1 \ x = x = x/ 1 Equations ■ Simple rules ( yz ) \ x = z \ ( y \ x ) and x/ ( zy ) = ( x/y ) /z ■ FEP Hypersequents x \ ( y/z ) = ( x \ y ) /z ■ Hyper-frames CE for HFL So, for N = P , ( P, ∨ , · , 1) acts on both sides on ( N, ∧ ) by \ and / . Relativizing to InFL FMP for InFL Thus, (( N, ∧ ) , \ , / ) becomes a ( P, ∨ , · , 1) -bimodule. This split DFL matches the notion of polarity . It also extend to � , � . FEP for IDFL CE for HDFL The bimodule can be viewed as a two-sorted algebra Relativising Conuclei ( P, ∨ , · , 1 , N, ∧ , \ , / ) . The absolutely free algebra for P = N generated by P 0 = N 0 = V ar (the set of propositional variables) gives the set of all formulas. Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 6 / 28

  7. Bi-modules Substructiral logics and residuated lattices Outline Residuated lattices x 1 = x = 1 x , ( xy ) z = x ( yz ) ■ Examples Bi-modules x ( y ∨ z ) = xy ∨ xz and ( y ∨ z ) x = yx ∨ zx ■ Formula hierarchy Submodules and nuclei So, if P is a residuated lattice, then ( P, ∨ , · , 1) is a semiring. [In the Lattice frames Residuated frames complete case, a quantale.] GN FL x \ ( y ∧ z ) = ( x \ y ) ∧ ( x \ z ) and ( y ∧ z ) /x = ( y/x ) ∧ ( z/x ) ■ Gentzen frames Compl - CE ( y ∨ z ) \ x = ( y \ x ) ∧ ( z \ x ) and x/ ( y ∨ z ) = ( x/y ) ∧ ( x/z ) ■ Frame applications 1 \ x = x = x/ 1 Equations ■ Simple rules ( yz ) \ x = z \ ( y \ x ) and x/ ( zy ) = ( x/y ) /z ■ FEP Hypersequents x \ ( y/z ) = ( x \ y ) /z ■ Hyper-frames CE for HFL So, for N = P , ( P, ∨ , · , 1) acts on both sides on ( N, ∧ ) by \ and / . Relativizing to InFL FMP for InFL Thus, (( N, ∧ ) , \ , / ) becomes a ( P, ∨ , · , 1) -bimodule. This split DFL matches the notion of polarity . It also extend to � , � . FEP for IDFL CE for HDFL The bimodule can be viewed as a two-sorted algebra Relativising Conuclei ( P, ∨ , · , 1 , N, ∧ , \ , / ) . The absolutely free algebra for P = N generated by P 0 = N 0 = V ar (the set of propositional variables) gives the set of all formulas. The steps of the generation process yield the substructural hierarchy . Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 6 / 28

  8. Formula hierarchy Substructiral logics and residuated lattices Outline Residuated lattices Examples ♣♣♣♣♣♣♣♣♣ ✻ ♣♣♣♣♣♣♣♣♣ ✻ Bi-modules The sets P n , N n of formulas are defined by: ■ Formula hierarchy (0) P 0 = N 0 = the set of variables Submodules and nuclei Lattice frames (P1) N n ⊆ P n +1 Residuated frames GN P 3 N 3 α, β ∈ P n +1 ⇒ α ∨ β, α · β, 1 ∈ P n +1 (P2) FL ✻ ■ ❅ � ✒ ✻ ❅ � P n ⊆ N n +1 (N1) Gentzen frames Compl - CE � ❅ α, β ∈ N n +1 ⇒ α ∧ β ∈ N n +1 (N2) Frame applications � ❅ Equations α ∈ P n +1 , β ∈ N n +1 ⇒ α \ β, β/α, 0 ∈ N n +1 (N3) Simple rules P 2 N 2 P n +1 = �N n � � , � ; N n +1 = �P n � � , P n +1 \ ,/ P n +1 FEP ■ ✻ ■ ❅ ❅ � � ✒ ✻ Hypersequents P n ⊆ P n +1 , N n ⊆ N n +1 , � P n = � N n = Fm Hyper-frames � ❅ ■ CE for HFL � ❅ P 1 -reduced: � � p i Relativizing to InFL ■ FMP for InFL P 1 N 1 DFL N 1 -reduced: � ( p 1 p 2 · · · p n \ r/q 1 q 2 · · · q m ) FEP for IDFL ✻ ■ ❅ � ✒ ✻ ■ ❅ � CE for HDFL � ❅ p 1 p 2 · · · p n q 1 q 2 · · · q m ≤ r Relativising � ❅ Conuclei Sequent: a 1 , a 2 , . . . , a n ⇒ a 0 ( a i ∈ Fm ) ■ P 0 N 0 A. Ciabattoni, NG, K. Terui. From axioms to analytic rules in nonclassical logics, Proceedings of LICS’08, 229-240, 2008. Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 7 / 28

  9. Submodules and nuclei Substructiral logics and residuated lattices Outline Residuated lattices Given a ( P, � , · , 1) -bimodule (( N, � ) , \ , / ) , each sub-bimodule is Examples Bi-modules defined by a � -closed subset that is also closed under the actions. Formula hierarchy Submodules and nuclei Lattice frames Residuated frames GN FL Gentzen frames Compl - CE Frame applications Equations Simple rules FEP Hypersequents Hyper-frames CE for HFL Relativizing to InFL FMP for InFL DFL FEP for IDFL CE for HDFL Relativising Conuclei Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 8 / 28

  10. Submodules and nuclei Substructiral logics and residuated lattices Outline Residuated lattices Given a ( P, � , · , 1) -bimodule (( N, � ) , \ , / ) , each sub-bimodule is Examples Bi-modules defined by a � -closed subset that is also closed under the actions. Formula hierarchy Submodules and nuclei Namely, it is defined by a nucleus : a closure operator γ on N such Lattice frames that p ∈ P, n ∈ N implies p \ n, n/p ∈ N . Residuated frames GN FL Gentzen frames Compl - CE Frame applications Equations Simple rules FEP Hypersequents Hyper-frames CE for HFL Relativizing to InFL FMP for InFL DFL FEP for IDFL CE for HDFL Relativising Conuclei Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 8 / 28

  11. Submodules and nuclei Substructiral logics and residuated lattices Outline Residuated lattices Given a ( P, � , · , 1) -bimodule (( N, � ) , \ , / ) , each sub-bimodule is Examples Bi-modules defined by a � -closed subset that is also closed under the actions. Formula hierarchy Submodules and nuclei Namely, it is defined by a nucleus : a closure operator γ on N such Lattice frames that p ∈ P, n ∈ N implies p \ n, n/p ∈ N . Residuated frames GN If P = N is the underlying set of a residuated lattice FL Gentzen frames A = � A, ∧ , ∨ , · , \ , /, 1 � , a nucleus is just a closure operator that Compl - CE Frame applications satisfies γ ( x ) · γ ( y ) ≤ γ ( x · y ) . (Cf. phase spaces.) Equations Simple rules FEP Hypersequents Hyper-frames CE for HFL Relativizing to InFL FMP for InFL DFL FEP for IDFL CE for HDFL Relativising Conuclei Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 8 / 28

  12. Submodules and nuclei Substructiral logics and residuated lattices Outline Residuated lattices Given a ( P, � , · , 1) -bimodule (( N, � ) , \ , / ) , each sub-bimodule is Examples Bi-modules defined by a � -closed subset that is also closed under the actions. Formula hierarchy Submodules and nuclei Namely, it is defined by a nucleus : a closure operator γ on N such Lattice frames that p ∈ P, n ∈ N implies p \ n, n/p ∈ N . Residuated frames GN If P = N is the underlying set of a residuated lattice FL Gentzen frames A = � A, ∧ , ∨ , · , \ , /, 1 � , a nucleus is just a closure operator that Compl - CE Frame applications satisfies γ ( x ) · γ ( y ) ≤ γ ( x · y ) . (Cf. phase spaces.) Equations Simple rules If we define A γ = { γ ( x ) : x ∈ A } , x ∨ γ y = γ ( x ∨ y ) and FEP x · γ y = γ ( x · y ) , Hypersequents Hyper-frames CE for HFL A γ = � A γ , ∧ , ∨ γ , · γ , \ , /, γ (1) � Relativizing to InFL FMP for InFL DFL is also a residuated lattice. FEP for IDFL CE for HDFL Relativising Conuclei Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 8 / 28

  13. Submodules and nuclei Substructiral logics and residuated lattices Outline Residuated lattices Given a ( P, � , · , 1) -bimodule (( N, � ) , \ , / ) , each sub-bimodule is Examples Bi-modules defined by a � -closed subset that is also closed under the actions. Formula hierarchy Submodules and nuclei Namely, it is defined by a nucleus : a closure operator γ on N such Lattice frames that p ∈ P, n ∈ N implies p \ n, n/p ∈ N . Residuated frames GN If P = N is the underlying set of a residuated lattice FL Gentzen frames A = � A, ∧ , ∨ , · , \ , /, 1 � , a nucleus is just a closure operator that Compl - CE Frame applications satisfies γ ( x ) · γ ( y ) ≤ γ ( x · y ) . (Cf. phase spaces.) Equations Simple rules If we define A γ = { γ ( x ) : x ∈ A } , x ∨ γ y = γ ( x ∨ y ) and FEP x · γ y = γ ( x · y ) , Hypersequents Hyper-frames CE for HFL A γ = � A γ , ∧ , ∨ γ , · γ , \ , /, γ (1) � Relativizing to InFL FMP for InFL DFL is also a residuated lattice. FEP for IDFL CE for HDFL Relativising All complete RLs arise as submodules of P ( M ) , where M is a Conuclei monoid, namely via nuclei on powersets (of monoids). Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 8 / 28

  14. Submodules and nuclei Substructiral logics and residuated lattices Outline Residuated lattices Given a ( P, � , · , 1) -bimodule (( N, � ) , \ , / ) , each sub-bimodule is Examples Bi-modules defined by a � -closed subset that is also closed under the actions. Formula hierarchy Submodules and nuclei Namely, it is defined by a nucleus : a closure operator γ on N such Lattice frames that p ∈ P, n ∈ N implies p \ n, n/p ∈ N . Residuated frames GN If P = N is the underlying set of a residuated lattice FL Gentzen frames A = � A, ∧ , ∨ , · , \ , /, 1 � , a nucleus is just a closure operator that Compl - CE Frame applications satisfies γ ( x ) · γ ( y ) ≤ γ ( x · y ) . (Cf. phase spaces.) Equations Simple rules If we define A γ = { γ ( x ) : x ∈ A } , x ∨ γ y = γ ( x ∨ y ) and FEP x · γ y = γ ( x · y ) , Hypersequents Hyper-frames CE for HFL A γ = � A γ , ∧ , ∨ γ , · γ , \ , /, γ (1) � Relativizing to InFL FMP for InFL DFL is also a residuated lattice. FEP for IDFL CE for HDFL Relativising All complete RLs arise as submodules of P ( M ) , where M is a Conuclei monoid, namely via nuclei on powersets (of monoids). (Each RL can be embedded into a complete one.) Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 8 / 28

  15. Submodules and nuclei Substructiral logics and residuated lattices Outline Residuated lattices Given a ( P, � , · , 1) -bimodule (( N, � ) , \ , / ) , each sub-bimodule is Examples Bi-modules defined by a � -closed subset that is also closed under the actions. Formula hierarchy Submodules and nuclei Namely, it is defined by a nucleus : a closure operator γ on N such Lattice frames that p ∈ P, n ∈ N implies p \ n, n/p ∈ N . Residuated frames GN If P = N is the underlying set of a residuated lattice FL Gentzen frames A = � A, ∧ , ∨ , · , \ , /, 1 � , a nucleus is just a closure operator that Compl - CE Frame applications satisfies γ ( x ) · γ ( y ) ≤ γ ( x · y ) . (Cf. phase spaces.) Equations Simple rules If we define A γ = { γ ( x ) : x ∈ A } , x ∨ γ y = γ ( x ∨ y ) and FEP x · γ y = γ ( x · y ) , Hypersequents Hyper-frames CE for HFL A γ = � A γ , ∧ , ∨ γ , · γ , \ , /, γ (1) � Relativizing to InFL FMP for InFL DFL is also a residuated lattice. FEP for IDFL CE for HDFL Relativising All complete RLs arise as submodules of P ( M ) , where M is a Conuclei monoid, namely via nuclei on powersets (of monoids). (Each RL can be embedded into a complete one.) Residuated frames arise from studying submodules of P ( M ) . Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 8 / 28

  16. Submodules and nuclei Substructiral logics and residuated lattices Outline Residuated lattices Given a ( P, � , · , 1) -bimodule (( N, � ) , \ , / ) , each sub-bimodule is Examples Bi-modules defined by a � -closed subset that is also closed under the actions. Formula hierarchy Submodules and nuclei Namely, it is defined by a nucleus : a closure operator γ on N such Lattice frames that p ∈ P, n ∈ N implies p \ n, n/p ∈ N . Residuated frames GN If P = N is the underlying set of a residuated lattice FL Gentzen frames A = � A, ∧ , ∨ , · , \ , /, 1 � , a nucleus is just a closure operator that Compl - CE Frame applications satisfies γ ( x ) · γ ( y ) ≤ γ ( x · y ) . (Cf. phase spaces.) Equations Simple rules If we define A γ = { γ ( x ) : x ∈ A } , x ∨ γ y = γ ( x ∨ y ) and FEP x · γ y = γ ( x · y ) , Hypersequents Hyper-frames CE for HFL A γ = � A γ , ∧ , ∨ γ , · γ , \ , /, γ (1) � Relativizing to InFL FMP for InFL DFL is also a residuated lattice. FEP for IDFL CE for HDFL Relativising All complete RLs arise as submodules of P ( M ) , where M is a Conuclei monoid, namely via nuclei on powersets (of monoids). (Each RL can be embedded into a complete one.) Residuated frames arise from studying submodules of P ( M ) .They form relational semantics for substructural logics Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 8 / 28

  17. Submodules and nuclei Substructiral logics and residuated lattices Outline Residuated lattices Given a ( P, � , · , 1) -bimodule (( N, � ) , \ , / ) , each sub-bimodule is Examples Bi-modules defined by a � -closed subset that is also closed under the actions. Formula hierarchy Submodules and nuclei Namely, it is defined by a nucleus : a closure operator γ on N such Lattice frames that p ∈ P, n ∈ N implies p \ n, n/p ∈ N . Residuated frames GN If P = N is the underlying set of a residuated lattice FL Gentzen frames A = � A, ∧ , ∨ , · , \ , /, 1 � , a nucleus is just a closure operator that Compl - CE Frame applications satisfies γ ( x ) · γ ( y ) ≤ γ ( x · y ) . (Cf. phase spaces.) Equations Simple rules If we define A γ = { γ ( x ) : x ∈ A } , x ∨ γ y = γ ( x ∨ y ) and FEP x · γ y = γ ( x · y ) , Hypersequents Hyper-frames CE for HFL A γ = � A γ , ∧ , ∨ γ , · γ , \ , /, γ (1) � Relativizing to InFL FMP for InFL DFL is also a residuated lattice. FEP for IDFL CE for HDFL Relativising All complete RLs arise as submodules of P ( M ) , where M is a Conuclei monoid, namely via nuclei on powersets (of monoids). (Each RL can be embedded into a complete one.) Residuated frames arise from studying submodules of P ( M ) .They form relational semantics for substructural logics and are the most important tool in Algebraic Proof Theory. Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 8 / 28

  18. Lattice frames Substructiral logics and residuated lattices Outline Residuated lattices A lattice frame is a structure F = ( L, R, N ) where L and R are sets Examples Bi-modules and N is a binary relation from L to R . Formula hierarchy Submodules and nuclei If A is a lattice, F A = ( A, A, ≤ ) is a lattice frame. Lattice frames Residuated frames GN FL Gentzen frames Compl - CE Frame applications Equations Simple rules FEP Hypersequents Hyper-frames CE for HFL Relativizing to InFL FMP for InFL DFL FEP for IDFL CE for HDFL Relativising Conuclei Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 9 / 28

  19. Lattice frames Substructiral logics and residuated lattices Outline Residuated lattices A lattice frame is a structure F = ( L, R, N ) where L and R are sets Examples Bi-modules and N is a binary relation from L to R . Formula hierarchy Submodules and nuclei If A is a lattice, F A = ( A, A, ≤ ) is a lattice frame. Lattice frames Residuated frames GN For X ⊆ L and Y ⊆ R we define FL Gentzen frames X ⊲ = { b ∈ R : x N b, for all x ∈ X } Compl - CE Frame applications Y ⊳ = { a ∈ L : a N y, for all y ∈ Y } Equations Simple rules FEP Hypersequents Hyper-frames CE for HFL Relativizing to InFL FMP for InFL DFL FEP for IDFL CE for HDFL Relativising Conuclei Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 9 / 28

  20. Lattice frames Substructiral logics and residuated lattices Outline Residuated lattices A lattice frame is a structure F = ( L, R, N ) where L and R are sets Examples Bi-modules and N is a binary relation from L to R . Formula hierarchy Submodules and nuclei If A is a lattice, F A = ( A, A, ≤ ) is a lattice frame. Lattice frames Residuated frames GN For X ⊆ L and Y ⊆ R we define FL Gentzen frames X ⊲ = { b ∈ R : x N b, for all x ∈ X } Compl - CE Frame applications Y ⊳ = { a ∈ L : a N y, for all y ∈ Y } Equations Simple rules FEP The maps ⊲ : P ( L ) → P ( R ) and ⊳ : P ( R ) → P ( L ) form a Galois Hypersequents Hyper-frames connection. The map γ N : P ( L ) → P ( L ) , where γ N ( X ) = X ⊲⊳ , is CE for HFL Relativizing to InFL a closure operator. FMP for InFL DFL FEP for IDFL CE for HDFL Relativising Conuclei Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 9 / 28

  21. Lattice frames Substructiral logics and residuated lattices Outline Residuated lattices A lattice frame is a structure F = ( L, R, N ) where L and R are sets Examples Bi-modules and N is a binary relation from L to R . Formula hierarchy Submodules and nuclei If A is a lattice, F A = ( A, A, ≤ ) is a lattice frame. Lattice frames Residuated frames GN For X ⊆ L and Y ⊆ R we define FL Gentzen frames X ⊲ = { b ∈ R : x N b, for all x ∈ X } Compl - CE Frame applications Y ⊳ = { a ∈ L : a N y, for all y ∈ Y } Equations Simple rules FEP The maps ⊲ : P ( L ) → P ( R ) and ⊳ : P ( R ) → P ( L ) form a Galois Hypersequents Hyper-frames connection. The map γ N : P ( L ) → P ( L ) , where γ N ( X ) = X ⊲⊳ , is CE for HFL Relativizing to InFL a closure operator. FMP for InFL DFL Lemma. If A = ( A, ∧ , ∨ ) is a lattice and γ is a cl.op. on L , then FEP for IDFL CE for HDFL ( γ [ A ] , ∧ , ∨ γ ) is a lattice. [ x ∨ γ y = γ ( x ∨ y ) .] Relativising Conuclei Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 9 / 28

  22. Lattice frames Substructiral logics and residuated lattices Outline Residuated lattices A lattice frame is a structure F = ( L, R, N ) where L and R are sets Examples Bi-modules and N is a binary relation from L to R . Formula hierarchy Submodules and nuclei If A is a lattice, F A = ( A, A, ≤ ) is a lattice frame. Lattice frames Residuated frames GN For X ⊆ L and Y ⊆ R we define FL Gentzen frames X ⊲ = { b ∈ R : x N b, for all x ∈ X } Compl - CE Frame applications Y ⊳ = { a ∈ L : a N y, for all y ∈ Y } Equations Simple rules FEP The maps ⊲ : P ( L ) → P ( R ) and ⊳ : P ( R ) → P ( L ) form a Galois Hypersequents Hyper-frames connection. The map γ N : P ( L ) → P ( L ) , where γ N ( X ) = X ⊲⊳ , is CE for HFL Relativizing to InFL a closure operator. FMP for InFL DFL Lemma. If A = ( A, ∧ , ∨ ) is a lattice and γ is a cl.op. on L , then FEP for IDFL CE for HDFL ( γ [ A ] , ∧ , ∨ γ ) is a lattice. [ x ∨ γ y = γ ( x ∨ y ) .] Relativising Conuclei Corollary. If F is a lattice frame then the Galois algebra F + = ( γ N [ P ( L )] , ∩ , ∪ γ N ) is a complete lattice. Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 9 / 28

  23. Lattice frames Substructiral logics and residuated lattices Outline Residuated lattices A lattice frame is a structure F = ( L, R, N ) where L and R are sets Examples Bi-modules and N is a binary relation from L to R . Formula hierarchy Submodules and nuclei If A is a lattice, F A = ( A, A, ≤ ) is a lattice frame. Lattice frames Residuated frames GN For X ⊆ L and Y ⊆ R we define FL Gentzen frames X ⊲ = { b ∈ R : x N b, for all x ∈ X } Compl - CE Frame applications Y ⊳ = { a ∈ L : a N y, for all y ∈ Y } Equations Simple rules FEP The maps ⊲ : P ( L ) → P ( R ) and ⊳ : P ( R ) → P ( L ) form a Galois Hypersequents Hyper-frames connection. The map γ N : P ( L ) → P ( L ) , where γ N ( X ) = X ⊲⊳ , is CE for HFL Relativizing to InFL a closure operator. FMP for InFL DFL Lemma. If A = ( A, ∧ , ∨ ) is a lattice and γ is a cl.op. on L , then FEP for IDFL CE for HDFL ( γ [ A ] , ∧ , ∨ γ ) is a lattice. [ x ∨ γ y = γ ( x ∨ y ) .] Relativising Conuclei Corollary. If F is a lattice frame then the Galois algebra F + = ( γ N [ P ( L )] , ∩ , ∪ γ N ) is a complete lattice. If A is a lattice, F + A is the Dedekind-MacNeille completion of A and x �→ { x } ⊳ is an embedding. Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 9 / 28

  24. Residuated frames Substructiral logics and residuated lattices Outline Residuated lattices A residuated frame is a structure F = ( L, R, N, ◦ , ε, � , � ) where Examples Bi-modules Formula hierarchy ( L, R, N ) is a lattice frame, ■ Submodules and nuclei ( L, ◦ , ε ) is a monoid and ■ Lattice frames � : L × R → R , � : R × L → R are such that Residuated frames ■ GN FL ( x ◦ y ) N w ⇔ y N ( x � w ) ⇔ x N ( w � y ) Gentzen frames Compl - CE Frame applications Equations Simple rules FEP Hypersequents Hyper-frames CE for HFL Relativizing to InFL FMP for InFL DFL FEP for IDFL CE for HDFL Relativising Conuclei Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 10 / 28

  25. Residuated frames Substructiral logics and residuated lattices Outline Residuated lattices A residuated frame is a structure F = ( L, R, N, ◦ , ε, � , � ) where Examples Bi-modules Formula hierarchy ( L, R, N ) is a lattice frame, ■ Submodules and nuclei ( L, ◦ , ε ) is a monoid and ■ Lattice frames � : L × R → R , � : R × L → R are such that Residuated frames ■ GN FL ( x ◦ y ) N w ⇔ y N ( x � w ) ⇔ x N ( w � y ) Gentzen frames Compl - CE Frame applications Theorem. If F is a frame, then γ N is a nucleus on P ( L, ◦ , 1) . Equations Simple rules FEP Hypersequents Hyper-frames CE for HFL Relativizing to InFL FMP for InFL DFL FEP for IDFL CE for HDFL Relativising Conuclei Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 10 / 28

  26. Residuated frames Substructiral logics and residuated lattices Outline Residuated lattices A residuated frame is a structure F = ( L, R, N, ◦ , ε, � , � ) where Examples Bi-modules Formula hierarchy ( L, R, N ) is a lattice frame, ■ Submodules and nuclei ( L, ◦ , ε ) is a monoid and ■ Lattice frames � : L × R → R , � : R × L → R are such that Residuated frames ■ GN FL ( x ◦ y ) N w ⇔ y N ( x � w ) ⇔ x N ( w � y ) Gentzen frames Compl - CE Frame applications Theorem. If F is a frame, then γ N is a nucleus on P ( L, ◦ , 1) . Then Equations Simple rules F + = P ( L, ◦ , ε ) γ N = ( P ( L ) γ N , ∩ , ∪ γ N , · γ N , \ , /, γ N ( ε )) is a FEP Hypersequents residuated lattice called the Galois algebra of F . Hyper-frames CE for HFL Relativizing to InFL FMP for InFL DFL FEP for IDFL CE for HDFL Relativising Conuclei Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 10 / 28

  27. Residuated frames Substructiral logics and residuated lattices Outline Residuated lattices A residuated frame is a structure F = ( L, R, N, ◦ , ε, � , � ) where Examples Bi-modules Formula hierarchy ( L, R, N ) is a lattice frame, ■ Submodules and nuclei ( L, ◦ , ε ) is a monoid and ■ Lattice frames � : L × R → R , � : R × L → R are such that Residuated frames ■ GN FL ( x ◦ y ) N w ⇔ y N ( x � w ) ⇔ x N ( w � y ) Gentzen frames Compl - CE Frame applications Theorem. If F is a frame, then γ N is a nucleus on P ( L, ◦ , 1) . Then Equations Simple rules F + = P ( L, ◦ , ε ) γ N = ( P ( L ) γ N , ∩ , ∪ γ N , · γ N , \ , /, γ N ( ε )) is a FEP Hypersequents residuated lattice called the Galois algebra of F . Hyper-frames CE for HFL If A is a RL, F A = ( A, A, ≤ , · , 1) is a residuated frame. Moreover, Relativizing to InFL for F A , x �→ { x } ⊳ is an embedding. FMP for InFL DFL FEP for IDFL CE for HDFL Relativising Conuclei Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 10 / 28

  28. Residuated frames Substructiral logics and residuated lattices Outline Residuated lattices A residuated frame is a structure F = ( L, R, N, ◦ , ε, � , � ) where Examples Bi-modules Formula hierarchy ( L, R, N ) is a lattice frame, ■ Submodules and nuclei ( L, ◦ , ε ) is a monoid and ■ Lattice frames � : L × R → R , � : R × L → R are such that Residuated frames ■ GN FL ( x ◦ y ) N w ⇔ y N ( x � w ) ⇔ x N ( w � y ) Gentzen frames Compl - CE Frame applications Theorem. If F is a frame, then γ N is a nucleus on P ( L, ◦ , 1) . Then Equations Simple rules F + = P ( L, ◦ , ε ) γ N = ( P ( L ) γ N , ∩ , ∪ γ N , · γ N , \ , /, γ N ( ε )) is a FEP Hypersequents residuated lattice called the Galois algebra of F . Hyper-frames CE for HFL If A is a RL, F A = ( A, A, ≤ , · , 1) is a residuated frame. Moreover, Relativizing to InFL for F A , x �→ { x } ⊳ is an embedding. FMP for InFL DFL Note: ( L, ◦ , ε ) acts on R via � and � (modulo N ). FEP for IDFL CE for HDFL Relativising Conuclei Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 10 / 28

  29. Residuated frames Substructiral logics and residuated lattices Outline Residuated lattices A residuated frame is a structure F = ( L, R, N, ◦ , ε, � , � ) where Examples Bi-modules Formula hierarchy ( L, R, N ) is a lattice frame, ■ Submodules and nuclei ( L, ◦ , ε ) is a monoid and ■ Lattice frames � : L × R → R , � : R × L → R are such that Residuated frames ■ GN FL ( x ◦ y ) N w ⇔ y N ( x � w ) ⇔ x N ( w � y ) Gentzen frames Compl - CE Frame applications Theorem. If F is a frame, then γ N is a nucleus on P ( L, ◦ , 1) . Then Equations Simple rules F + = P ( L, ◦ , ε ) γ N = ( P ( L ) γ N , ∩ , ∪ γ N , · γ N , \ , /, γ N ( ε )) is a FEP Hypersequents residuated lattice called the Galois algebra of F . Hyper-frames CE for HFL If A is a RL, F A = ( A, A, ≤ , · , 1) is a residuated frame. Moreover, Relativizing to InFL for F A , x �→ { x } ⊳ is an embedding. FMP for InFL DFL Note: ( L, ◦ , ε ) acts on R via � and � (modulo N ). A frame is freely FEP for IDFL CE for HDFL generated by B , if ( L, ◦ , 1) is the free monoid B ∗ and R is (bijective Relativising to) B ∗ × B × B ∗ ≡ S B ∗ × B . Conuclei Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 10 / 28

  30. Residuated frames Substructiral logics and residuated lattices Outline Residuated lattices A residuated frame is a structure F = ( L, R, N, ◦ , ε, � , � ) where Examples Bi-modules Formula hierarchy ( L, R, N ) is a lattice frame, ■ Submodules and nuclei ( L, ◦ , ε ) is a monoid and ■ Lattice frames � : L × R → R , � : R × L → R are such that Residuated frames ■ GN FL ( x ◦ y ) N w ⇔ y N ( x � w ) ⇔ x N ( w � y ) Gentzen frames Compl - CE Frame applications Theorem. If F is a frame, then γ N is a nucleus on P ( L, ◦ , 1) . Then Equations Simple rules F + = P ( L, ◦ , ε ) γ N = ( P ( L ) γ N , ∩ , ∪ γ N , · γ N , \ , /, γ N ( ε )) is a FEP Hypersequents residuated lattice called the Galois algebra of F . Hyper-frames CE for HFL If A is a RL, F A = ( A, A, ≤ , · , 1) is a residuated frame. Moreover, Relativizing to InFL for F A , x �→ { x } ⊳ is an embedding. FMP for InFL DFL Note: ( L, ◦ , ε ) acts on R via � and � (modulo N ). A frame is freely FEP for IDFL CE for HDFL generated by B , if ( L, ◦ , 1) is the free monoid B ∗ and R is (bijective Relativising to) B ∗ × B × B ∗ ≡ S B ∗ × B . (Given a monoid L = ( L, ◦ , ε ) , S L Conuclei denotes the sections (unary linear polynomials) of L .) N. Galatos and P. Jipsen. Residuated frames and applications to decidability, to appear in the Transactions of the AMS. Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 10 / 28

  31. GN Substructiral logics and residuated lattices Outline Residuated lattices xNa aNz Examples (CUT) aNa (Id) Bi-modules xNz Formula hierarchy Submodules and nuclei Lattice frames Residuated frames GN FL Gentzen frames Compl - CE Frame applications Equations Simple rules FEP Hypersequents Hyper-frames CE for HFL Relativizing to InFL FMP for InFL DFL FEP for IDFL CE for HDFL Relativising Conuclei Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 11 / 28

  32. GN Substructiral logics and residuated lattices Outline Residuated lattices xNa aNz Examples (CUT) aNa (Id) Bi-modules xNz Formula hierarchy Submodules and nuclei aNz bNz xNa xNb ( ∨ L) xNa ∨ b ( ∨ R ℓ ) xNa ∨ b ( ∨ R r ) Lattice frames a ∨ bNz Residuated frames GN FL Gentzen frames Compl - CE Frame applications Equations Simple rules FEP Hypersequents Hyper-frames CE for HFL Relativizing to InFL FMP for InFL DFL FEP for IDFL CE for HDFL Relativising Conuclei Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 11 / 28

  33. GN Substructiral logics and residuated lattices Outline Residuated lattices xNa aNz Examples (CUT) aNa (Id) Bi-modules xNz Formula hierarchy Submodules and nuclei aNz bNz xNa xNb ( ∨ L) xNa ∨ b ( ∨ R ℓ ) xNa ∨ b ( ∨ R r ) Lattice frames a ∨ bNz Residuated frames GN aNz bNz xNa xNb FL a ∧ bNz ( ∧ L ℓ ) a ∧ bNz ( ∧ L r ) ( ∧ R) Gentzen frames xNa ∧ b Compl - CE Frame applications Equations Simple rules FEP Hypersequents Hyper-frames CE for HFL Relativizing to InFL FMP for InFL DFL FEP for IDFL CE for HDFL Relativising Conuclei Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 11 / 28

  34. GN Substructiral logics and residuated lattices Outline Residuated lattices xNa aNz Examples (CUT) aNa (Id) Bi-modules xNz Formula hierarchy Submodules and nuclei aNz bNz xNa xNb ( ∨ L) xNa ∨ b ( ∨ R ℓ ) xNa ∨ b ( ∨ R r ) Lattice frames a ∨ bNz Residuated frames GN aNz bNz xNa xNb FL a ∧ bNz ( ∧ L ℓ ) a ∧ bNz ( ∧ L r ) ( ∧ R) Gentzen frames xNa ∧ b Compl - CE Frame applications xNa yNb a ◦ bNz εNz Equations a · bNz ( · L) x ◦ yNa · b ( · R) 1 Nz ( 1 L) εN 1 ( 1 R) Simple rules FEP Hypersequents Hyper-frames CE for HFL Relativizing to InFL FMP for InFL DFL FEP for IDFL CE for HDFL Relativising Conuclei Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 11 / 28

  35. GN Substructiral logics and residuated lattices Outline Residuated lattices xNa aNz Examples (CUT) aNa (Id) Bi-modules xNz Formula hierarchy Submodules and nuclei aNz bNz xNa xNb ( ∨ L) xNa ∨ b ( ∨ R ℓ ) xNa ∨ b ( ∨ R r ) Lattice frames a ∨ bNz Residuated frames GN aNz bNz xNa xNb FL a ∧ bNz ( ∧ L ℓ ) a ∧ bNz ( ∧ L r ) ( ∧ R) Gentzen frames xNa ∧ b Compl - CE Frame applications xNa yNb a ◦ bNz εNz Equations a · bNz ( · L) x ◦ yNa · b ( · R) 1 Nz ( 1 L) εN 1 ( 1 R) Simple rules FEP Hypersequents xNa � b xNa bNz Hyper-frames a \ bNx � z ( \ L) ( \ R) CE for HFL xNa \ b Relativizing to InFL FMP for InFL DFL FEP for IDFL CE for HDFL Relativising Conuclei Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 11 / 28

  36. GN Substructiral logics and residuated lattices Outline Residuated lattices xNa aNz Examples (CUT) aNa (Id) Bi-modules xNz Formula hierarchy Submodules and nuclei aNz bNz xNa xNb ( ∨ L) xNa ∨ b ( ∨ R ℓ ) xNa ∨ b ( ∨ R r ) Lattice frames a ∨ bNz Residuated frames GN aNz bNz xNa xNb FL a ∧ bNz ( ∧ L ℓ ) a ∧ bNz ( ∧ L r ) ( ∧ R) Gentzen frames xNa ∧ b Compl - CE Frame applications xNa yNb a ◦ bNz εNz Equations a · bNz ( · L) x ◦ yNa · b ( · R) 1 Nz ( 1 L) εN 1 ( 1 R) Simple rules FEP Hypersequents xNa � b xNa bNz Hyper-frames a \ bNx � z ( \ L) ( \ R) CE for HFL xNa \ b Relativizing to InFL FMP for InFL xNb � a xNa bNz DFL b/aNz � x ( / L) ( / R) FEP for IDFL xNb/a CE for HDFL Relativising Conuclei Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 11 / 28

  37. GN Substructiral logics and residuated lattices Outline Residuated lattices xNa aNz Examples (CUT) aNa (Id) Bi-modules xNz Formula hierarchy Submodules and nuclei aNz bNz xNa xNb ( ∨ L) xNa ∨ b ( ∨ R ℓ ) xNa ∨ b ( ∨ R r ) Lattice frames a ∨ bNz Residuated frames GN aNz bNz xNa xNb FL a ∧ bNz ( ∧ L ℓ ) a ∧ bNz ( ∧ L r ) ( ∧ R) Gentzen frames xNa ∧ b Compl - CE Frame applications xNa yNb a ◦ bNz εNz Equations a · bNz ( · L) x ◦ yNa · b ( · R) 1 Nz ( 1 L) εN 1 ( 1 R) Simple rules FEP Hypersequents xNa � b xNa bNz Hyper-frames a \ bNx � z ( \ L) ( \ R) CE for HFL xNa \ b Relativizing to InFL FMP for InFL xNb � a xNa bNz DFL b/aNz � x ( / L) ( / R) FEP for IDFL xNb/a CE for HDFL Relativising Conuclei xNa bNz x ◦ ( a \ b ) Nz Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 11 / 28

  38. GN Substructiral logics and residuated lattices Outline Residuated lattices xNa aNz Examples (CUT) aNa (Id) Bi-modules xNz Formula hierarchy Submodules and nuclei aNz bNz xNa xNb ( ∨ L) xNa ∨ b ( ∨ R ℓ ) xNa ∨ b ( ∨ R r ) Lattice frames a ∨ bNz Residuated frames GN aNz bNz xNa xNb FL a ∧ bNz ( ∧ L ℓ ) a ∧ bNz ( ∧ L r ) ( ∧ R) Gentzen frames xNa ∧ b Compl - CE Frame applications xNa yNb a ◦ bNz εNz Equations a · bNz ( · L) x ◦ yNa · b ( · R) 1 Nz ( 1 L) εN 1 ( 1 R) Simple rules FEP Hypersequents xNa � b xNa bNz Hyper-frames a \ bNx � z ( \ L) ( \ R) CE for HFL xNa \ b Relativizing to InFL FMP for InFL xNb � a xNa bNz DFL b/aNz � x ( / L) ( / R) FEP for IDFL xNb/a CE for HDFL Relativising xNa bN ( v � c � u ) Conuclei xNa bNz x ◦ ( a \ b ) Nz x ◦ ( a \ b ) N ( v � c � u ) Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 11 / 28

  39. GN Substructiral logics and residuated lattices Outline Residuated lattices xNa aNz Examples (CUT) aNa (Id) Bi-modules xNz Formula hierarchy Submodules and nuclei aNz bNz xNa xNb ( ∨ L) xNa ∨ b ( ∨ R ℓ ) xNa ∨ b ( ∨ R r ) Lattice frames a ∨ bNz Residuated frames GN aNz bNz xNa xNb FL a ∧ bNz ( ∧ L ℓ ) a ∧ bNz ( ∧ L r ) ( ∧ R) Gentzen frames xNa ∧ b Compl - CE Frame applications xNa yNb a ◦ bNz εNz Equations a · bNz ( · L) x ◦ yNa · b ( · R) 1 Nz ( 1 L) εN 1 ( 1 R) Simple rules FEP Hypersequents xNa � b xNa bNz Hyper-frames a \ bNx � z ( \ L) ( \ R) CE for HFL xNa \ b Relativizing to InFL FMP for InFL xNb � a xNa bNz DFL b/aNz � x ( / L) ( / R) FEP for IDFL xNb/a CE for HDFL Relativising xNa bN ( v � c � u ) Conuclei v ◦ b ◦ uNc xNa bNz xNa x ◦ ( a \ b ) Nz x ◦ ( a \ b ) N ( v � c � u ) v ◦ x ◦ ( a \ b ) ◦ uNc So, we get the sequent calculus FL , for a, b, c ∈ Fm , x, y, u, v ∈ Fm ∗ , z ∈ S L × Fm . Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 11 / 28

  40. FL Substructiral logics and residuated lattices Outline Residuated lattices x ⇒ a y ◦ a ◦ z ⇒ c Examples (cut) a ⇒ a (Id) y ◦ x ◦ z ⇒ c Bi-modules Formula hierarchy Submodules and nuclei y ◦ a ◦ z ⇒ c y ◦ b ◦ z ⇒ c x ⇒ a x ⇒ b Lattice frames y ◦ a ∧ b ◦ z ⇒ c ( ∧ L ℓ ) y ◦ a ∧ b ◦ z ⇒ c ( ∧ L r ) ( ∧ R) Residuated frames x ⇒ a ∧ b GN FL y ◦ a ◦ z ⇒ c y ◦ b ◦ z ⇒ c x ⇒ a x ⇒ b Gentzen frames ( ∨ L) x ⇒ a ∨ b ( ∨ R ℓ ) x ⇒ a ∨ b ( ∨ R r ) y ◦ a ∨ b ◦ z ⇒ c Compl - CE Frame applications Equations x ⇒ a y ◦ b ◦ z ⇒ c a ◦ x ⇒ b Simple rules y ◦ x ◦ ( a \ b ) ◦ z ⇒ c ( \ L) x ⇒ a \ b ( \ R) FEP Hypersequents Hyper-frames x ⇒ a y ◦ b ◦ z ⇒ c x ◦ a ⇒ b CE for HFL y ◦ ( b/a ) ◦ x ◦ z ⇒ c ( / L) x ⇒ b/a ( / R) Relativizing to InFL FMP for InFL DFL y ◦ a ◦ b ◦ z ⇒ c x ⇒ a y ⇒ b FEP for IDFL y ◦ a · b ◦ z ⇒ c ( · L) ( · R) CE for HDFL x ◦ y ⇒ a · b Relativising Conuclei y ◦ z ⇒ a y ◦ 1 ◦ z ⇒ a ( 1 L) ε ⇒ 1 ( 1 R) where a, b, c ∈ Fm , x, y, z ∈ Fm ∗ . Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 12 / 28

  41. Gentzen frames Substructiral logics and residuated lattices Outline Residuated lattices F FL is the free frame generated by the formulas Fm ( L = Fm ∗ , Examples Bi-modules R = S L × Fm ), whith x N ( u, a ) iff ⊢ FL u ( x ) ⇒ a . Formula hierarchy Submodules and nuclei Lattice frames Residuated frames GN FL Gentzen frames Compl - CE Frame applications Equations Simple rules FEP Hypersequents Hyper-frames CE for HFL Relativizing to InFL FMP for InFL DFL FEP for IDFL CE for HDFL Relativising Conuclei Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 13 / 28

  42. Gentzen frames Substructiral logics and residuated lattices Outline Residuated lattices F FL is the free frame generated by the formulas Fm ( L = Fm ∗ , Examples Bi-modules R = S L × Fm ), whith x N ( u, a ) iff ⊢ FL u ( x ) ⇒ a . Formula hierarchy Submodules and nuclei The following properties hold for F A , F FL (and F A , B , later): Lattice frames Residuated frames 1. F is a residuated frame (freely) generated by B GN FL 2. B is a (partial) algebra of the same type, ( B = A , Fm ) Gentzen frames Compl - CE 3. N satisfies GN , for all a, b ∈ B , x, y ∈ L , z ∈ R . Frame applications Equations Simple rules FEP Hypersequents Hyper-frames CE for HFL Relativizing to InFL FMP for InFL DFL FEP for IDFL CE for HDFL Relativising Conuclei Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 13 / 28

  43. Gentzen frames Substructiral logics and residuated lattices Outline Residuated lattices F FL is the free frame generated by the formulas Fm ( L = Fm ∗ , Examples Bi-modules R = S L × Fm ), whith x N ( u, a ) iff ⊢ FL u ( x ) ⇒ a . Formula hierarchy Submodules and nuclei The following properties hold for F A , F FL (and F A , B , later): Lattice frames Residuated frames 1. F is a residuated frame (freely) generated by B GN FL 2. B is a (partial) algebra of the same type, ( B = A , Fm ) Gentzen frames Compl - CE 3. N satisfies GN , for all a, b ∈ B , x, y ∈ L , z ∈ R . Frame applications Equations We call such pairs ( F , B ) Gentzen frames . A cut-free Gentzen frame Simple rules FEP is not assumed to satisfy the (CUT)-rule. Hypersequents Hyper-frames CE for HFL Relativizing to InFL FMP for InFL DFL FEP for IDFL CE for HDFL Relativising Conuclei Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 13 / 28

  44. Gentzen frames Substructiral logics and residuated lattices Outline Residuated lattices F FL is the free frame generated by the formulas Fm ( L = Fm ∗ , Examples Bi-modules R = S L × Fm ), whith x N ( u, a ) iff ⊢ FL u ( x ) ⇒ a . Formula hierarchy Submodules and nuclei The following properties hold for F A , F FL (and F A , B , later): Lattice frames Residuated frames 1. F is a residuated frame (freely) generated by B GN FL 2. B is a (partial) algebra of the same type, ( B = A , Fm ) Gentzen frames Compl - CE 3. N satisfies GN , for all a, b ∈ B , x, y ∈ L , z ∈ R . Frame applications Equations We call such pairs ( F , B ) Gentzen frames . A cut-free Gentzen frame Simple rules FEP is not assumed to satisfy the (CUT)-rule. Hypersequents Hyper-frames Theorem. (NG-Jipsen) Given a Gentzen frame ( F , B ) , the map CE for HFL {} ⊳ : B → F + , b �→ { b } ⊳ is a (partial) homomorphism. Relativizing to InFL FMP for InFL (Namely, if a, b ∈ B and a • b ∈ B ( • is a connective) then DFL { a • B b } ⊳ = { a } ⊳ • F + { b } ⊳ ). FEP for IDFL CE for HDFL Relativising Conuclei Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 13 / 28

  45. Gentzen frames Substructiral logics and residuated lattices Outline Residuated lattices F FL is the free frame generated by the formulas Fm ( L = Fm ∗ , Examples Bi-modules R = S L × Fm ), whith x N ( u, a ) iff ⊢ FL u ( x ) ⇒ a . Formula hierarchy Submodules and nuclei The following properties hold for F A , F FL (and F A , B , later): Lattice frames Residuated frames 1. F is a residuated frame (freely) generated by B GN FL 2. B is a (partial) algebra of the same type, ( B = A , Fm ) Gentzen frames Compl - CE 3. N satisfies GN , for all a, b ∈ B , x, y ∈ L , z ∈ R . Frame applications Equations We call such pairs ( F , B ) Gentzen frames . A cut-free Gentzen frame Simple rules FEP is not assumed to satisfy the (CUT)-rule. Hypersequents Hyper-frames Theorem. (NG-Jipsen) Given a Gentzen frame ( F , B ) , the map CE for HFL {} ⊳ : B → F + , b �→ { b } ⊳ is a (partial) homomorphism. Relativizing to InFL FMP for InFL (Namely, if a, b ∈ B and a • b ∈ B ( • is a connective) then DFL { a • B b } ⊳ = { a } ⊳ • F + { b } ⊳ ). FEP for IDFL CE for HDFL Relativising For cut-free Genzten frames, we get only a quasihomomorphism . Conuclei a • B b ∈ { a } ⊳ • F + { b } ⊳ ⊆ { a • B b } ⊳ . Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 13 / 28

  46. Completeness - Cut elimination Substructiral logics and residuated lattices Outline Residuated lattices f : Fm L → F + be the For every homomorphism f : Fm → B , let ¯ Examples Bi-modules f ( p ) = { f ( p ) } ⊳ ( p : variable.) homomorphism that extends ¯ Formula hierarchy Submodules and nuclei Corollary. If ( F , B ) is a cf Gentzen frame, for every homomorphism Lattice frames f : Fm → B , we have f ( a ) ∈ ¯ Residuated frames f ( a ) ⊆ { f ( a ) } ⊳ . If we have (CUT), GN then ¯ f ( a ) = ↓ f ( a ) . FL Gentzen frames Compl - CE We define F | = x ⇒ c by f ( x ) N f ( c ) , for all f . Frame applications = x · ≤ c , then F FL | Equations Theorem. If F + FL | = x ⇒ c . Simple rules Idea: For f : Fm → B , f ( x ) ∈ ¯ f ( x ) ⊆ ¯ f ( c ) ⊆ { f ( c ) } ⊳ , so FEP Hypersequents f ( x ) N f ( c ) . Hyper-frames CE for HFL Corollary. FL is complete with respect to F + FL . Relativizing to InFL FMP for InFL Corollary (CE). FL and FL f prove the same sequents. DFL FEP for IDFL CE for HDFL Theorem . (Ciabattoni-NG-Terui) For axioms in N 2 , the extension of Relativising Conuclei FL is equivalent to one that admits (modular, infinitary) cut elimination iff the corresponding variety is closed under (MacNeille) completions iff the axiom is acyclic . Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 14 / 28

  47. Frame applications Substructiral logics and residuated lattices Outline Residuated lattices DM-completion ■ Examples Bi-modules Completeness of the calculus ■ Formula hierarchy Submodules and nuclei Cut elimination ■ Lattice frames Residuated frames Finite model property ■ GN FL Finite embeddability property ■ Gentzen frames Compl - CE (Generalized super-)amalgamation property (Transferable ■ Frame applications injections, Congruence extension property) Equations Simple rules (Craig) Interpolation property FEP ■ Hypersequents Disjunction property ■ Hyper-frames CE for HFL Strong separation ■ Relativizing to InFL FMP for InFL Stability under linear structural rules/equations over {∨ , · , 1 } . ■ DFL FEP for IDFL CE for HDFL NG and H. Ono, APAL. Relativising NG and P. Jipsen, TAMS. Conuclei NG and P. Jipsen, manuscript. A. Ciabattoni, NG and K. Terui, APAL. NG and K. Terui, manuscript. Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 15 / 28

  48. Equations Substructiral logics and residuated lattices Outline Residuated lattices Idea: Express equations over {∨ , · , 1 } at the frame level. Examples Bi-modules Formula hierarchy Submodules and nuclei Lattice frames Residuated frames GN FL Gentzen frames Compl - CE Frame applications Equations Simple rules FEP Hypersequents Hyper-frames CE for HFL Relativizing to InFL FMP for InFL DFL FEP for IDFL CE for HDFL Relativising Conuclei Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 16 / 28

  49. Equations Substructiral logics and residuated lattices Outline Residuated lattices Idea: Express equations over {∨ , · , 1 } at the frame level. Examples Bi-modules Formula hierarchy For an equation ε over {∨ , · , 1 } we distribute products over joins to Submodules and nuclei get s 1 ∨ · · · ∨ s m = t 1 ∨ · · · ∨ t n . s i , t j : monoid terms. Lattice frames Residuated frames GN FL Gentzen frames Compl - CE Frame applications Equations Simple rules FEP Hypersequents Hyper-frames CE for HFL Relativizing to InFL FMP for InFL DFL FEP for IDFL CE for HDFL Relativising Conuclei Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 16 / 28

  50. Equations Substructiral logics and residuated lattices Outline Residuated lattices Idea: Express equations over {∨ , · , 1 } at the frame level. Examples Bi-modules Formula hierarchy For an equation ε over {∨ , · , 1 } we distribute products over joins to Submodules and nuclei get s 1 ∨ · · · ∨ s m = t 1 ∨ · · · ∨ t n . s i , t j : monoid terms. Lattice frames Residuated frames GN s 1 ∨ · · · ∨ s m ≤ t 1 ∨ · · · ∨ t n and t 1 ∨ · · · ∨ t n ≤ s 1 ∨ · · · ∨ s m . FL Gentzen frames Compl - CE Frame applications Equations Simple rules FEP Hypersequents Hyper-frames CE for HFL Relativizing to InFL FMP for InFL DFL FEP for IDFL CE for HDFL Relativising Conuclei Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 16 / 28

  51. Equations Substructiral logics and residuated lattices Outline Residuated lattices Idea: Express equations over {∨ , · , 1 } at the frame level. Examples Bi-modules Formula hierarchy For an equation ε over {∨ , · , 1 } we distribute products over joins to Submodules and nuclei get s 1 ∨ · · · ∨ s m = t 1 ∨ · · · ∨ t n . s i , t j : monoid terms. Lattice frames Residuated frames GN s 1 ∨ · · · ∨ s m ≤ t 1 ∨ · · · ∨ t n and t 1 ∨ · · · ∨ t n ≤ s 1 ∨ · · · ∨ s m . FL Gentzen frames The first is equivalent to: &( s j ≤ t 1 ∨ · · · ∨ t n ) . Compl - CE Frame applications Equations Simple rules FEP Hypersequents Hyper-frames CE for HFL Relativizing to InFL FMP for InFL DFL FEP for IDFL CE for HDFL Relativising Conuclei Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 16 / 28

  52. Equations Substructiral logics and residuated lattices Outline Residuated lattices Idea: Express equations over {∨ , · , 1 } at the frame level. Examples Bi-modules Formula hierarchy For an equation ε over {∨ , · , 1 } we distribute products over joins to Submodules and nuclei get s 1 ∨ · · · ∨ s m = t 1 ∨ · · · ∨ t n . s i , t j : monoid terms. Lattice frames Residuated frames GN s 1 ∨ · · · ∨ s m ≤ t 1 ∨ · · · ∨ t n and t 1 ∨ · · · ∨ t n ≤ s 1 ∨ · · · ∨ s m . FL Gentzen frames The first is equivalent to: &( s j ≤ t 1 ∨ · · · ∨ t n ) . Compl - CE Frame applications Equations We proceed by example: x 2 y ≤ xy ∨ yx Simple rules FEP Hypersequents Hyper-frames CE for HFL Relativizing to InFL FMP for InFL DFL FEP for IDFL CE for HDFL Relativising Conuclei Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 16 / 28

  53. Equations Substructiral logics and residuated lattices Outline Residuated lattices Idea: Express equations over {∨ , · , 1 } at the frame level. Examples Bi-modules Formula hierarchy For an equation ε over {∨ , · , 1 } we distribute products over joins to Submodules and nuclei get s 1 ∨ · · · ∨ s m = t 1 ∨ · · · ∨ t n . s i , t j : monoid terms. Lattice frames Residuated frames GN s 1 ∨ · · · ∨ s m ≤ t 1 ∨ · · · ∨ t n and t 1 ∨ · · · ∨ t n ≤ s 1 ∨ · · · ∨ s m . FL Gentzen frames The first is equivalent to: &( s j ≤ t 1 ∨ · · · ∨ t n ) . Compl - CE Frame applications Equations We proceed by example: x 2 y ≤ xy ∨ yx Simple rules FEP ( x 1 ∨ x 2 ) 2 y ≤ ( x 1 ∨ x 2 ) y ∨ y ( x 1 ∨ x 2 ) Hypersequents Hyper-frames CE for HFL Relativizing to InFL FMP for InFL DFL FEP for IDFL CE for HDFL Relativising Conuclei Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 16 / 28

  54. Equations Substructiral logics and residuated lattices Outline Residuated lattices Idea: Express equations over {∨ , · , 1 } at the frame level. Examples Bi-modules Formula hierarchy For an equation ε over {∨ , · , 1 } we distribute products over joins to Submodules and nuclei get s 1 ∨ · · · ∨ s m = t 1 ∨ · · · ∨ t n . s i , t j : monoid terms. Lattice frames Residuated frames GN s 1 ∨ · · · ∨ s m ≤ t 1 ∨ · · · ∨ t n and t 1 ∨ · · · ∨ t n ≤ s 1 ∨ · · · ∨ s m . FL Gentzen frames The first is equivalent to: &( s j ≤ t 1 ∨ · · · ∨ t n ) . Compl - CE Frame applications Equations We proceed by example: x 2 y ≤ xy ∨ yx Simple rules FEP ( x 1 ∨ x 2 ) 2 y ≤ ( x 1 ∨ x 2 ) y ∨ y ( x 1 ∨ x 2 ) Hypersequents Hyper-frames CE for HFL x 2 1 y ∨ x 1 x 2 y ∨ x 2 x 1 y ∨ x 2 2 y ≤ x 1 y ∨ x 2 y ∨ yx 1 ∨ yx 2 Relativizing to InFL FMP for InFL DFL FEP for IDFL CE for HDFL Relativising Conuclei Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 16 / 28

  55. Equations Substructiral logics and residuated lattices Outline Residuated lattices Idea: Express equations over {∨ , · , 1 } at the frame level. Examples Bi-modules Formula hierarchy For an equation ε over {∨ , · , 1 } we distribute products over joins to Submodules and nuclei get s 1 ∨ · · · ∨ s m = t 1 ∨ · · · ∨ t n . s i , t j : monoid terms. Lattice frames Residuated frames GN s 1 ∨ · · · ∨ s m ≤ t 1 ∨ · · · ∨ t n and t 1 ∨ · · · ∨ t n ≤ s 1 ∨ · · · ∨ s m . FL Gentzen frames The first is equivalent to: &( s j ≤ t 1 ∨ · · · ∨ t n ) . Compl - CE Frame applications Equations We proceed by example: x 2 y ≤ xy ∨ yx Simple rules FEP ( x 1 ∨ x 2 ) 2 y ≤ ( x 1 ∨ x 2 ) y ∨ y ( x 1 ∨ x 2 ) Hypersequents Hyper-frames CE for HFL x 2 1 y ∨ x 1 x 2 y ∨ x 2 x 1 y ∨ x 2 2 y ≤ x 1 y ∨ x 2 y ∨ yx 1 ∨ yx 2 Relativizing to InFL FMP for InFL x 1 x 2 y ≤ x 1 y ∨ x 2 y ∨ yx 1 ∨ yx 2 DFL FEP for IDFL CE for HDFL Relativising Conuclei Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 16 / 28

  56. Equations Substructiral logics and residuated lattices Outline Residuated lattices Idea: Express equations over {∨ , · , 1 } at the frame level. Examples Bi-modules Formula hierarchy For an equation ε over {∨ , · , 1 } we distribute products over joins to Submodules and nuclei get s 1 ∨ · · · ∨ s m = t 1 ∨ · · · ∨ t n . s i , t j : monoid terms. Lattice frames Residuated frames GN s 1 ∨ · · · ∨ s m ≤ t 1 ∨ · · · ∨ t n and t 1 ∨ · · · ∨ t n ≤ s 1 ∨ · · · ∨ s m . FL Gentzen frames The first is equivalent to: &( s j ≤ t 1 ∨ · · · ∨ t n ) . Compl - CE Frame applications Equations We proceed by example: x 2 y ≤ xy ∨ yx Simple rules FEP ( x 1 ∨ x 2 ) 2 y ≤ ( x 1 ∨ x 2 ) y ∨ y ( x 1 ∨ x 2 ) Hypersequents Hyper-frames CE for HFL x 2 1 y ∨ x 1 x 2 y ∨ x 2 x 1 y ∨ x 2 2 y ≤ x 1 y ∨ x 2 y ∨ yx 1 ∨ yx 2 Relativizing to InFL FMP for InFL x 1 x 2 y ≤ x 1 y ∨ x 2 y ∨ yx 1 ∨ yx 2 DFL FEP for IDFL CE for HDFL x 1 y ≤ z x 2 y ≤ z yx 1 ≤ z yx 2 ≤ z Relativising Conuclei x 1 x 2 y ≤ z Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 16 / 28

  57. Equations Substructiral logics and residuated lattices Outline Residuated lattices Idea: Express equations over {∨ , · , 1 } at the frame level. Examples Bi-modules Formula hierarchy For an equation ε over {∨ , · , 1 } we distribute products over joins to Submodules and nuclei get s 1 ∨ · · · ∨ s m = t 1 ∨ · · · ∨ t n . s i , t j : monoid terms. Lattice frames Residuated frames GN s 1 ∨ · · · ∨ s m ≤ t 1 ∨ · · · ∨ t n and t 1 ∨ · · · ∨ t n ≤ s 1 ∨ · · · ∨ s m . FL Gentzen frames The first is equivalent to: &( s j ≤ t 1 ∨ · · · ∨ t n ) . Compl - CE Frame applications Equations We proceed by example: x 2 y ≤ xy ∨ yx Simple rules FEP ( x 1 ∨ x 2 ) 2 y ≤ ( x 1 ∨ x 2 ) y ∨ y ( x 1 ∨ x 2 ) Hypersequents Hyper-frames CE for HFL x 2 1 y ∨ x 1 x 2 y ∨ x 2 x 1 y ∨ x 2 2 y ≤ x 1 y ∨ x 2 y ∨ yx 1 ∨ yx 2 Relativizing to InFL FMP for InFL x 1 x 2 y ≤ x 1 y ∨ x 2 y ∨ yx 1 ∨ yx 2 DFL FEP for IDFL CE for HDFL x 1 y ≤ z x 2 y ≤ z yx 1 ≤ z yx 2 ≤ z Relativising Conuclei x 1 x 2 y ≤ z x 1 ◦ y N z x 2 ◦ y N z y ◦ x 1 N z y ◦ x 2 N z R ( ε ) x 1 ◦ x 2 ◦ y N z Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 16 / 28

  58. Simple rules Substructiral logics and residuated lattices Outline Residuated lattices In the context of ( F FL , Fm ) , R ( ε ) takes the form Examples Bi-modules Formula hierarchy x ◦ t 1 ◦ y ⇒ a · · · x ◦ t n ◦ y ⇒ a Submodules and nuclei ( R ( ε )) Lattice frames x ◦ t 0 ◦ y ⇒ a Residuated frames GN FL We call such equations and rules simple. Gentzen frames Compl - CE Theorem. Let ( F , B ) be a cf Gentzen frame and let ε be a Frame applications {∨ , · , 1 } -equation. Then ( F , B ) satisfies R ( ε ) iff F + satisfies ε . Equations Simple rules FEP Theorem. All extensions of FL by simple rules enjoy cut elimination. Hypersequents Hyper-frames K. Terui. Which structural rules admit cut elimination? An algebraic CE for HFL Relativizing to InFL criterion. J. Symbolic Logic 72 (2007), no. 3, 738-754. FMP for InFL DFL N. Galatos and H. Ono. Cut elimination and strong separation for FEP for IDFL CE for HDFL non-associative substructural logics, APAL 161(9) (2010), Relativising Conuclei 1097–1133. N. Galatos and P. Jipsen. Residuated frames and applications to decidability, to appear in the Transactions of the AMS. Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 17 / 28

  59. FEP Substructiral logics and residuated lattices Outline Residuated lattices Theorem. Every variety V of integral RL’s ( x ≤ 1 ) axiomatized by Examples Bi-modules equartions over {∨ , · , 1 } has the finite embeddability property (FEP) , Formula hierarchy Submodules and nuclei namely for every A ∈ V , every finite partial subalgebra B of A can Lattice frames be (partially) embedded in a finite D ∈ V . Residuated frames GN FL The frame F A , B is generated by B ( L is the submonoid of A Gentzen frames generated by B , R = S L × B ) with x N ( u, b ) iff u ( x ) ≤ A b . Compl - CE Frame applications Equations Simple rules FEP Hypersequents Hyper-frames CE for HFL Relativizing to InFL FMP for InFL DFL FEP for IDFL CE for HDFL Relativising Conuclei Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 18 / 28

  60. FEP Substructiral logics and residuated lattices Outline Residuated lattices Theorem. Every variety V of integral RL’s ( x ≤ 1 ) axiomatized by Examples Bi-modules equartions over {∨ , · , 1 } has the finite embeddability property (FEP) , Formula hierarchy Submodules and nuclei namely for every A ∈ V , every finite partial subalgebra B of A can Lattice frames be (partially) embedded in a finite D ∈ V . Residuated frames GN FL The frame F A , B is generated by B ( L is the submonoid of A Gentzen frames generated by B , R = S L × B ) with x N ( u, b ) iff u ( x ) ≤ A b . Then Compl - CE Frame applications Equations F + A , B ∈ V ■ Simple rules FEP A , B via { } ⊳ : B → F + B embeds in F + Hypersequents ■ Hyper-frames CE for HFL F + A , B is finite ■ Relativizing to InFL FMP for InFL DFL N. Galatos and P. Jipsen. Residuated frames and applications to FEP for IDFL decidability, to appear in the Transactions of the AMS. CE for HDFL Relativising Conuclei Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 18 / 28

  61. Hypersequents Substructiral logics and residuated lattices Outline Residuated lattices FL sequents stem from N 1 -normal formulas. FL supports the Examples Bi-modules analysis of simple structural rules, which correspond to N 2 -equations. Formula hierarchy Submodules and nuclei Lattice frames Residuated frames GN FL Gentzen frames Compl - CE Frame applications Equations Simple rules FEP Hypersequents Hyper-frames CE for HFL Relativizing to InFL FMP for InFL DFL FEP for IDFL CE for HDFL Relativising Conuclei Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 19 / 28

  62. Hypersequents Substructiral logics and residuated lattices Outline Residuated lattices FL sequents stem from N 1 -normal formulas. FL supports the Examples Bi-modules analysis of simple structural rules, which correspond to N 2 -equations. Formula hierarchy Submodules and nuclei To handle P 3 -equations, we define hypersequents, based on Lattice frames P 2 -normal formulas: ( x 1 . . . x n → x 0 ) ∨ ( y 1 . . . y n → y 0 ) ∨ . . . . Residuated frames GN FL Gentzen frames Compl - CE Frame applications Equations Simple rules FEP Hypersequents Hyper-frames CE for HFL Relativizing to InFL FMP for InFL DFL FEP for IDFL CE for HDFL Relativising Conuclei Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 19 / 28

  63. Hypersequents Substructiral logics and residuated lattices Outline Residuated lattices FL sequents stem from N 1 -normal formulas. FL supports the Examples Bi-modules analysis of simple structural rules, which correspond to N 2 -equations. Formula hierarchy Submodules and nuclei To handle P 3 -equations, we define hypersequents, based on Lattice frames P 2 -normal formulas: ( x 1 . . . x n → x 0 ) ∨ ( y 1 . . . y n → y 0 ) ∨ . . . . Residuated frames GN A hypersequent is a multiset s 1 | · · · | s m of sequents s i . FL Gentzen frames Compl - CE Frame applications Equations Simple rules FEP Hypersequents Hyper-frames CE for HFL Relativizing to InFL FMP for InFL DFL FEP for IDFL CE for HDFL Relativising Conuclei Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 19 / 28

  64. Hypersequents Substructiral logics and residuated lattices Outline Residuated lattices FL sequents stem from N 1 -normal formulas. FL supports the Examples Bi-modules analysis of simple structural rules, which correspond to N 2 -equations. Formula hierarchy Submodules and nuclei To handle P 3 -equations, we define hypersequents, based on Lattice frames P 2 -normal formulas: ( x 1 . . . x n → x 0 ) ∨ ( y 1 . . . y n → y 0 ) ∨ . . . . Residuated frames GN A hypersequent is a multiset s 1 | · · · | s m of sequents s i . FL Gentzen frames Compl - CE For every rule Frame applications s 1 s 2 Equations s Simple rules FEP Hypersequents of FL , the system HFL is defined to contain the rule Hyper-frames CE for HFL H | s 1 H | s 2 Relativizing to InFL FMP for InFL H | s DFL FEP for IDFL CE for HDFL where H is a (meta)variable for hyprsequents. Relativising Conuclei Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 19 / 28

  65. Hypersequents Substructiral logics and residuated lattices Outline Residuated lattices FL sequents stem from N 1 -normal formulas. FL supports the Examples Bi-modules analysis of simple structural rules, which correspond to N 2 -equations. Formula hierarchy Submodules and nuclei To handle P 3 -equations, we define hypersequents, based on Lattice frames P 2 -normal formulas: ( x 1 . . . x n → x 0 ) ∨ ( y 1 . . . y n → y 0 ) ∨ . . . . Residuated frames GN A hypersequent is a multiset s 1 | · · · | s m of sequents s i . FL Gentzen frames Compl - CE For every rule Frame applications s 1 s 2 Equations s Simple rules FEP Hypersequents of FL , the system HFL is defined to contain the rule Hyper-frames CE for HFL H | s 1 H | s 2 Relativizing to InFL FMP for InFL H | s DFL FEP for IDFL CE for HDFL where H is a (meta)variable for hyprsequents. A hyperstructural rule Relativising Conuclei is of the form H | s ′ H | s ′ H | s ′ . . . n 1 2 H | s 1 | · · · | s m Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 19 / 28

  66. Hyper-frames Substructiral logics and residuated lattices Outline Residuated lattices A hyperresiduated frame H = ( L, R, ⊢ , ◦ , ε, � , � , ǫ ) is defined by Examples Bi-modules ⊢⊆ H = ( L × R ) ∗ . We write ⊢ h instead of h ∈⊢ . ■ Formula hierarchy Submodules and nuclei ( L, ◦ , ε ) is a monoid and ǫ ∈ R . ■ Lattice frames Residuated frames For all x, y ∈ L , z ∈ R , h ∈ H , ■ GN FL Gentzen frames ⊢ ( x ◦ y, z ) | h ⇔ ⊢ ( y, x � z ) | h ⇔ ⊢ ( x, z � y ) | h. Compl - CE Frame applications Equations ⊢ h implies ⊢ ( x, y ) | h for any ( x, y ) ∈ L × R . ■ Simple rules FEP ⊢ ( x, y ) | ( x, y ) | h implies ⊢ ( x, y ) | h for any ( x, y ) ∈ L × R . ■ Hypersequents Hyper-frames We define r ( H ) = ( L × H, R × H, N, • , ( ε ; ∅ ) , ( ǫ ; ∅ )) , where CE for HFL Relativizing to InFL H = ( L × R ) ∗ . Then r ( H ) is a residuated frame. We define FMP for InFL H + = r ( H ) + . The hyper-MacNeille completion of a residuated DFL FEP for IDFL lattice A is H + A . CE for HDFL Relativising Conuclei ( x ; h 1 ) • ( y ; h 2 ) = ( x ◦ y ; h 1 | h 2 ) ( x ; h 1 ) � ( z ; h 2 ) = ( x � z ; h 1 | h 2 ) ( z ; h 2 ) � ( x ; h 1 ) = ( z � x ; h 1 | h 2 ) ⇔ ⊢ ( x, z ) | h 1 | h 2 . ( x ; h 1 ) N ( z ; h 2 ) Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 20 / 28

  67. CE for HFL Substructiral logics and residuated lattices Outline Residuated lattices Example. Based on HFL we define a hyperresiduated frame Examples Bi-modules H HFL = ( W, W ′ , ⊢ , ◦ , ε, ǫ ) , where Formula hierarchy Submodules and nuclei Lattice frames ⊢ s 1 | . . . | s n ⇐ ⇒ ⊢ HFL s 1 | · · · | s n Residuated frames GN FL Gentzen frames Compl - CE Frame applications Using the cut-free version of this frame, we can prove cut elimination Equations Simple rules for HFL . FEP Hypersequents Hyper-frames The Dedekind-MacNeille and the hyper-Dedekind-MacNeille CE for HFL completions for N 2 and P 3 correspond in a strong way to modular Relativizing to InFL FMP for InFL cut elimination and to conservativity of the infinitary logic. DFL FEP for IDFL CE for HDFL A. Ciabattoni, NG, K. Terui. From axioms to analytic rules in Relativising nonclassical logics, Proceedings of LICS’08, 229-240, 2008. Conuclei A. Ciabattoni, NG, K. Terui. Algebraic proof theory for substructural logics: cut elimination and completions, to appear in APAL. Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 21 / 28

  68. Relativizing to InFL Substructiral logics and residuated lattices Outline Residuated lattices Recall that 0 is of type N n , hence ∼ x, − x : P n → N n . Examples Bi-modules Formula hierarchy Submodules and nuclei Lattice frames Residuated frames GN FL Gentzen frames Compl - CE Frame applications Equations Simple rules FEP Hypersequents Hyper-frames CE for HFL Relativizing to InFL FMP for InFL DFL FEP for IDFL CE for HDFL Relativising Conuclei Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 22 / 28

  69. Relativizing to InFL Substructiral logics and residuated lattices Outline Residuated lattices Recall that 0 is of type N n , hence ∼ x, − x : P n → N n . Examples Bi-modules If we add a new type to negations ∼ x, − x : N n → P n , then we arrive Formula hierarchy Submodules and nuclei at a new notion of sequent (multiple conclusion). The operations at Lattice frames Residuated frames the frame level corresponding to the negations are denoted by {} ∼ GN and {} − . FL Gentzen frames Compl - CE x ◦ y ⇒ z x ◦ y ⇒ z Frame applications x ⇒ z ◦ y − ( − ) y ⇒ x ∼ ◦ z ( ∼ ) Equations Simple rules FEP Hypersequents Hyper-frames CE for HFL Relativizing to InFL FMP for InFL DFL FEP for IDFL CE for HDFL Relativising Conuclei Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 22 / 28

  70. Relativizing to InFL Substructiral logics and residuated lattices Outline Residuated lattices Recall that 0 is of type N n , hence ∼ x, − x : P n → N n . Examples Bi-modules If we add a new type to negations ∼ x, − x : N n → P n , then we arrive Formula hierarchy Submodules and nuclei at a new notion of sequent (multiple conclusion). The operations at Lattice frames Residuated frames the frame level corresponding to the negations are denoted by {} ∼ GN and {} − . FL Gentzen frames Compl - CE x ◦ y ⇒ z x ◦ y ⇒ z Frame applications x ⇒ z ◦ y − ( − ) y ⇒ x ∼ ◦ z ( ∼ ) Equations Simple rules FEP Hypersequents An involutive (residuated) frame is a structure of the form Hyper-frames CE for HFL F = ( L = R, N, ◦ , ε, ∼ , − ) , where Relativizing to InFL FMP for InFL ( L, ◦ , ε, ∼ , − ) is weakly bi-involutive monoid, namely DFL ■ FEP for IDFL ( L, ◦ , ε ) is a monoid ◆ CE for HDFL x ∼− = x = x −∼ Relativising ◆ Conuclei ( y ∼ ◦ x ∼ ) − = ( y − ◦ x − ) ∼ [=: x ⊕ y ] ◆ x ◦ y N z iff y N x ∼ ⊕ z iff x N z ⊕ y − , for all x, y, z ∈ L ■ Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 22 / 28

  71. FMP for InFL Substructiral logics and residuated lattices Outline Residuated lattices Theorem The system InFL has cut elimination, FMP (and is Examples Bi-modules decidable). Its simple extensions all have cut elimination. Formula hierarchy Submodules and nuclei Lattice frames N. Galatos and P. Jipsen. Residuated frames and applications to Residuated frames decidability, to appear in the Transactions of the AMS. GN FL Gentzen frames Compl - CE HInFL e has cut elimination (via a syntactic argument, for now). Frame applications Equations A. Ciabattoni, L. Strassburger and K. Terui. Expanding the realm of Simple rules FEP systematic proof theory. Hypersequents Hyper-frames CE for HFL Relativizing to InFL FMP for InFL DFL FEP for IDFL CE for HDFL Relativising Conuclei Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 23 / 28

  72. DFL Substructiral logics and residuated lattices Outline Residuated lattices Recall that ∧ : N n × N n → N n . Examples Bi-modules Formula hierarchy Submodules and nuclei Lattice frames Residuated frames GN FL Gentzen frames Compl - CE Frame applications Equations Simple rules FEP Hypersequents Hyper-frames CE for HFL Relativizing to InFL FMP for InFL DFL FEP for IDFL CE for HDFL Relativising Conuclei Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 24 / 28

  73. DFL Substructiral logics and residuated lattices Outline Residuated lattices Recall that ∧ : N n × N n → N n . If we add ∧ : P n × P n → P n as a Examples Bi-modules new type, then we arrive at a new notion of sequent. The operation Formula hierarchy Submodules and nuclei at the frame level corresponding to ∧ is denoted by � ∧ . We obtain Lattice frames distributive sequents (Giambrone, Brady), and the calculus DFL . Residuated frames GN FL Gentzen frames Compl - CE Frame applications Equations Simple rules FEP Hypersequents Hyper-frames CE for HFL Relativizing to InFL FMP for InFL DFL FEP for IDFL CE for HDFL Relativising Conuclei Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 24 / 28

  74. DFL Substructiral logics and residuated lattices Outline Residuated lattices Recall that ∧ : N n × N n → N n . If we add ∧ : P n × P n → P n as a Examples Bi-modules new type, then we arrive at a new notion of sequent. The operation Formula hierarchy Submodules and nuclei at the frame level corresponding to ∧ is denoted by � ∧ . We obtain Lattice frames distributive sequents (Giambrone, Brady), and the calculus DFL . Residuated frames GN A distributive residuated frame ( dr-frame ) is a structure FL Gentzen frames F = ( L, R, N, ◦ , � , � , ε, � ∧ , � � , � � ) , where ( L, ◦ , ε ) is a monoid Compl - CE Frame applications ( L, � ∧ ) is a semilattice, N ⊆ L × R and Equations Simple rules ∧ : L 2 → L , � , � ◦ , � � : L × R → L , � , � � : R × L → R , FEP ■ Hypersequents x ◦ yNz iff xNz � y iff yNx � z . ■ Hyper-frames CE for HFL x � ∧ yNz iff xNz � � y iff yNx � � z . ■ Relativizing to InFL FMP for InFL xNw implies x � ∧ yNw ; and ■ DFL FEP for IDFL CE for HDFL Theorem. If F is a dr-frame then the Galois algebra Relativising F + = ( P ( L ) , ∩ , ∪ , ◦ , \ , /, 1) γ N is a distributive residuated lattice. Conuclei Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 24 / 28

  75. DFL Substructiral logics and residuated lattices Outline Residuated lattices Recall that ∧ : N n × N n → N n . If we add ∧ : P n × P n → P n as a Examples Bi-modules new type, then we arrive at a new notion of sequent. The operation Formula hierarchy Submodules and nuclei at the frame level corresponding to ∧ is denoted by � ∧ . We obtain Lattice frames distributive sequents (Giambrone, Brady), and the calculus DFL . Residuated frames GN A distributive residuated frame ( dr-frame ) is a structure FL Gentzen frames F = ( L, R, N, ◦ , � , � , ε, � ∧ , � � , � � ) , where ( L, ◦ , ε ) is a monoid Compl - CE Frame applications ( L, � ∧ ) is a semilattice, N ⊆ L × R and Equations Simple rules ∧ : L 2 → L , � , � ◦ , � � : L × R → L , � , � � : R × L → R , FEP ■ Hypersequents x ◦ yNz iff xNz � y iff yNx � z . ■ Hyper-frames CE for HFL x � ∧ yNz iff xNz � � y iff yNx � � z . ■ Relativizing to InFL FMP for InFL xNw implies x � ∧ yNw ; and ■ DFL FEP for IDFL CE for HDFL Theorem. If F is a dr-frame then the Galois algebra Relativising F + = ( P ( L ) , ∩ , ∪ , ◦ , \ , /, 1) γ N is a distributive residuated lattice. Conuclei DFL has cut elimination (also, all of its extensions with {∧ , ∨ , · , 1 } -equations/rules). It also has the FMP. N. Galatos and P. Jipsen. Cut elimination and the finite model property for distributive FL, manuscript. Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 24 / 28

  76. FEP for IDFL Substructiral logics and residuated lattices Outline Residuated lattices Let V be a subvariety of DIRL axiomatized over {∨ , ∧ , · , 1 } . To Examples Bi-modules establish the FEP for V , for every A in V and B a finite partial Formula hierarchy subalgebra of A , we construct an algebra D = F + Submodules and nuclei A , B such that Lattice frames Residuated frames F + GN A , B ∈ V ■ FL Gentzen frames B embeds in F + ■ Compl - CE A , B Frame applications Equations F + A , B is finite ■ Simple rules FEP Hypersequents Hyper-frames CE for HFL Relativizing to InFL FMP for InFL DFL FEP for IDFL CE for HDFL Relativising Conuclei Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 25 / 28

  77. FEP for IDFL Substructiral logics and residuated lattices Outline Residuated lattices Let V be a subvariety of DIRL axiomatized over {∨ , ∧ , · , 1 } . To Examples Bi-modules establish the FEP for V , for every A in V and B a finite partial Formula hierarchy subalgebra of A , we construct an algebra D = F + Submodules and nuclei A , B such that Lattice frames Residuated frames F + GN A , B ∈ V ■ FL Gentzen frames B embeds in F + ■ Compl - CE A , B Frame applications Equations F + A , B is finite ■ Simple rules FEP Hypersequents F + A , B is defined by taking ( L, ◦ , � ∧ , 1) to be the {· , ∧ , 1 } -subreduct Hyper-frames CE for HFL of A generated by B , R = S L × B and x N ( u, b ) iff u ( x ) ≤ A b . Relativizing to InFL FMP for InFL DFL FEP for IDFL CE for HDFL Relativising Conuclei Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 25 / 28

  78. FEP for IDFL Substructiral logics and residuated lattices Outline Residuated lattices Let V be a subvariety of DIRL axiomatized over {∨ , ∧ , · , 1 } . To Examples Bi-modules establish the FEP for V , for every A in V and B a finite partial Formula hierarchy subalgebra of A , we construct an algebra D = F + Submodules and nuclei A , B such that Lattice frames Residuated frames F + GN A , B ∈ V ■ FL Gentzen frames B embeds in F + ■ Compl - CE A , B Frame applications Equations F + A , B is finite ■ Simple rules FEP Hypersequents F + A , B is defined by taking ( L, ◦ , � ∧ , 1) to be the {· , ∧ , 1 } -subreduct Hyper-frames CE for HFL of A generated by B , R = S L × B and x N ( u, b ) iff u ( x ) ≤ A b . Relativizing to InFL FMP for InFL DFL Theorem. (NG) Every subvariety of DIRL axiomatized over FEP for IDFL CE for HDFL {∨ , ∧ , · , 1 } has the FEP. Relativising Conuclei Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 25 / 28

  79. CE for HDFL Substructiral logics and residuated lattices Outline Residuated lattices We consider distributive hypersequents , namely multisets Examples Bi-modules s 1 | · · · | s m , where s i ’s are distributive sequents. Formula hierarchy Submodules and nuclei Lattice frames Residuated frames GN FL Gentzen frames Compl - CE Frame applications Equations Simple rules FEP Hypersequents Hyper-frames CE for HFL Relativizing to InFL FMP for InFL DFL FEP for IDFL CE for HDFL Relativising Conuclei Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 26 / 28

  80. CE for HDFL Substructiral logics and residuated lattices Outline Residuated lattices We consider distributive hypersequents , namely multisets Examples Bi-modules s 1 | · · · | s m , where s i ’s are distributive sequents. We also consider Formula hierarchy Submodules and nuclei the Gentzen-style system HDFL . Lattice frames Residuated frames GN FL Gentzen frames Compl - CE Frame applications Equations Simple rules FEP Hypersequents Hyper-frames CE for HFL Relativizing to InFL FMP for InFL DFL FEP for IDFL CE for HDFL Relativising Conuclei Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 26 / 28

  81. CE for HDFL Substructiral logics and residuated lattices Outline Residuated lattices We consider distributive hypersequents , namely multisets Examples Bi-modules s 1 | · · · | s m , where s i ’s are distributive sequents. We also consider Formula hierarchy Submodules and nuclei the Gentzen-style system HDFL . Lattice frames Residuated frames We define distributive hyper-frames by allowing the relation ⊢ to GN FL ‘residuate’ with respect to both ◦ and � ∧ . Gentzen frames Compl - CE Frame applications Equations Simple rules FEP Hypersequents Hyper-frames CE for HFL Relativizing to InFL FMP for InFL DFL FEP for IDFL CE for HDFL Relativising Conuclei Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 26 / 28

  82. CE for HDFL Substructiral logics and residuated lattices Outline Residuated lattices We consider distributive hypersequents , namely multisets Examples Bi-modules s 1 | · · · | s m , where s i ’s are distributive sequents. We also consider Formula hierarchy Submodules and nuclei the Gentzen-style system HDFL . Lattice frames Residuated frames We define distributive hyper-frames by allowing the relation ⊢ to GN FL ‘residuate’ with respect to both ◦ and � ∧ . Gentzen frames Compl - CE Theorem. (Ciabbatoni-NG-Terui) The system HDFL has cut Frame applications Equations elimination. The same holds for all extensions by simple distributive Simple rules hyper-ryles corresponding to P 3 -equations on the distributive FEP Hypersequents hierarchy. Hyper-frames CE for HFL Relativizing to InFL FMP for InFL DFL FEP for IDFL CE for HDFL Relativising Conuclei Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 26 / 28

  83. CE for HDFL Substructiral logics and residuated lattices Outline Residuated lattices We consider distributive hypersequents , namely multisets Examples Bi-modules s 1 | · · · | s m , where s i ’s are distributive sequents. We also consider Formula hierarchy Submodules and nuclei the Gentzen-style system HDFL . Lattice frames Residuated frames We define distributive hyper-frames by allowing the relation ⊢ to GN FL ‘residuate’ with respect to both ◦ and � ∧ . Gentzen frames Compl - CE Theorem. (Ciabbatoni-NG-Terui) The system HDFL has cut Frame applications Equations elimination. The same holds for all extensions by simple distributive Simple rules hyper-ryles corresponding to P 3 -equations on the distributive FEP Hypersequents hierarchy. Hyper-frames CE for HFL In the process we discover a distributive hyper-MacNeille completion. Relativizing to InFL FMP for InFL DFL FEP for IDFL CE for HDFL Relativising Conuclei Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 26 / 28

  84. Relativising Substructiral logics and residuated lattices Outline Residuated lattices We can pick any monotone term (like ∧ ) and give a new type to it. Examples Bi-modules Formula hierarchy Submodules and nuclei Lattice frames Residuated frames GN FL Gentzen frames Compl - CE Frame applications Equations Simple rules FEP Hypersequents Hyper-frames CE for HFL Relativizing to InFL FMP for InFL DFL FEP for IDFL CE for HDFL Relativising Conuclei Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 27 / 28

  85. Relativising Substructiral logics and residuated lattices Outline Residuated lattices We can pick any monotone term (like ∧ ) and give a new type to it. Examples Bi-modules Formula hierarchy At the frame level we introduce a new metalogical connective and we Submodules and nuclei add a rule/condition that introduces the new term on the left from Lattice frames Residuated frames the new connective. GN FL Gentzen frames Compl - CE Frame applications Equations Simple rules FEP Hypersequents Hyper-frames CE for HFL Relativizing to InFL FMP for InFL DFL FEP for IDFL CE for HDFL Relativising Conuclei Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 27 / 28

  86. Relativising Substructiral logics and residuated lattices Outline Residuated lattices We can pick any monotone term (like ∧ ) and give a new type to it. Examples Bi-modules Formula hierarchy At the frame level we introduce a new metalogical connective and we Submodules and nuclei add a rule/condition that introduces the new term on the left from Lattice frames Residuated frames the new connective. GN FL Gentzen frames We can either write the rule ( ∨ L) with respect to the old context, or Compl - CE with respect to the new context and assume distribution of the new Frame applications Equations term over join. In the latter case, we work with a subvariety Simple rules (distributive RL in our example). FEP Hypersequents Hyper-frames CE for HFL Relativizing to InFL FMP for InFL DFL FEP for IDFL CE for HDFL Relativising Conuclei Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 27 / 28

  87. Relativising Substructiral logics and residuated lattices Outline Residuated lattices We can pick any monotone term (like ∧ ) and give a new type to it. Examples Bi-modules Formula hierarchy At the frame level we introduce a new metalogical connective and we Submodules and nuclei add a rule/condition that introduces the new term on the left from Lattice frames Residuated frames the new connective. GN FL Gentzen frames We can either write the rule ( ∨ L) with respect to the old context, or Compl - CE with respect to the new context and assume distribution of the new Frame applications Equations term over join. In the latter case, we work with a subvariety Simple rules (distributive RL in our example). FEP Hypersequents Hyper-frames (Ciabbatoni-NG-Terui) Then cut elimination holds, and the hierarchy CE for HFL Relativizing to InFL extends on this basis. Hence, we can define hypersequents over the FMP for InFL new structure etc. DFL FEP for IDFL CE for HDFL Relativising Conuclei Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 27 / 28

  88. Relativising Substructiral logics and residuated lattices Outline Residuated lattices We can pick any monotone term (like ∧ ) and give a new type to it. Examples Bi-modules Formula hierarchy At the frame level we introduce a new metalogical connective and we Submodules and nuclei add a rule/condition that introduces the new term on the left from Lattice frames Residuated frames the new connective. GN FL Gentzen frames We can either write the rule ( ∨ L) with respect to the old context, or Compl - CE with respect to the new context and assume distribution of the new Frame applications Equations term over join. In the latter case, we work with a subvariety Simple rules (distributive RL in our example). FEP Hypersequents Hyper-frames (Ciabbatoni-NG-Terui) Then cut elimination holds, and the hierarchy CE for HFL Relativizing to InFL extends on this basis. Hence, we can define hypersequents over the FMP for InFL new structure etc. DFL FEP for IDFL CE for HDFL This can lead to a plethora of completions. Relativising Conuclei Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 27 / 28

  89. Relativising Substructiral logics and residuated lattices Outline Residuated lattices We can pick any monotone term (like ∧ ) and give a new type to it. Examples Bi-modules Formula hierarchy At the frame level we introduce a new metalogical connective and we Submodules and nuclei add a rule/condition that introduces the new term on the left from Lattice frames Residuated frames the new connective. GN FL Gentzen frames We can either write the rule ( ∨ L) with respect to the old context, or Compl - CE with respect to the new context and assume distribution of the new Frame applications Equations term over join. In the latter case, we work with a subvariety Simple rules (distributive RL in our example). FEP Hypersequents Hyper-frames (Ciabbatoni-NG-Terui) Then cut elimination holds, and the hierarchy CE for HFL Relativizing to InFL extends on this basis. Hence, we can define hypersequents over the FMP for InFL new structure etc. DFL FEP for IDFL CE for HDFL This can lead to a plethora of completions. Relativising Conuclei We are working on the multiple conclusion case. Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 27 / 28

  90. Conuclei Substructiral logics and residuated lattices Outline Residuated lattices Given a ( P, � , · , 1) -bimodule (( N, � ) , \ , / ) , each homomorphic Examples Bi-modules image is defined (up to isomorphism) by a co-nucleus : an interior Formula hierarchy Submodules and nuclei operator σ over N that satisfies p \ σ ( n ) = σ ( p \ n ) . Lattice frames Residuated frames GN FL Gentzen frames Compl - CE Frame applications Equations Simple rules FEP Hypersequents Hyper-frames CE for HFL Relativizing to InFL FMP for InFL DFL FEP for IDFL CE for HDFL Relativising Conuclei Nikolaos Galatos, TACL’11, Marseille, July 2011 Relativising the substructural hierarchy – 28 / 28

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