Developments on higher levels of the substructural hierarchy Nick Galatos University of Denver ngalatos@du.edu August, 2013 Nick Galatos, BLAST, August 2013 Developments on higher levels of the substructural hierarchy – 1 / 20
Residuated lattices Residuated lattices Lattice frames Formula hierarchy Residuated frames Substructural logics are axiomatic extensions of FL : Gentzen’s GN FL sequent calculus for intuitionistic logic minus the structural rules of Frame applications exchange, contraction and weakening. Hypersequents Frame constructions Hyper-frames Examples Extra structure Hyper and PUFs Extensions Diagrams ALG and MV ℓ -pregroups Embedding theorems Maps on a chain Nick Galatos, BLAST, August 2013 Developments on higher levels of the substructural hierarchy – 2 / 20
Residuated lattices Residuated lattices Lattice frames Formula hierarchy Residuated frames Substructural logics are axiomatic extensions of FL : Gentzen’s GN FL sequent calculus for intuitionistic logic minus the structural rules of Frame applications exchange, contraction and weakening. Hypersequents Frame constructions Hyper-frames A residuated lattice , or residuated lattice-ordered monoid , is an Examples algebra A = ( A, ∧ , ∨ , · , \ , /, 1) such that Extra structure Hyper and PUFs ( A, ∧ , ∨ ) is a lattice, Extensions ■ Diagrams ( A, · , 1) is a monoid and ■ ALG and MV ℓ -pregroups for all a, b, c ∈ A , ■ Embedding theorems Maps on a chain ab ≤ c ⇔ b ≤ a \ c ⇔ a ≤ c/b. Nick Galatos, BLAST, August 2013 Developments on higher levels of the substructural hierarchy – 2 / 20
Residuated lattices Residuated lattices Lattice frames Formula hierarchy Residuated frames Substructural logics are axiomatic extensions of FL : Gentzen’s GN FL sequent calculus for intuitionistic logic minus the structural rules of Frame applications exchange, contraction and weakening. Hypersequents Frame constructions Hyper-frames A residuated lattice , or residuated lattice-ordered monoid , is an Examples algebra A = ( A, ∧ , ∨ , · , \ , /, 1) such that Extra structure Hyper and PUFs ( A, ∧ , ∨ ) is a lattice, Extensions ■ Diagrams ( A, · , 1) is a monoid and ■ ALG and MV ℓ -pregroups for all a, b, c ∈ A , ■ Embedding theorems Maps on a chain ab ≤ c ⇔ b ≤ a \ c ⇔ a ≤ c/b. Residuated lattice appear in both - Algebra : Lattice-ordered groups, relation algebras, ideals of a ring, quantales. - Logic : As models of various logics: Classical, intuitionistic, many-valued, linear, relevance logic. N. Galatos, P. Jipsen, T. Kowalski and H. Ono. Residuated Lattices: an algebraic glimpse at substructural logics, Studies in Logics and the Foundations of Mathematics, Elsevier, 2007. Nick Galatos, BLAST, August 2013 Developments on higher levels of the substructural hierarchy – 2 / 20
Lattice frames Residuated lattices Lattice frames Formula hierarchy Residuated frames GN FL Frame applications c a c d d ≤ a d c Hypersequents Frame constructions a × × Hyper-frames a a c b b Examples × × b Extra structure Hyper and PUFs × c Extensions Diagrams ALG and MV We obtain an oriented bipartite graph/ lattice frame F = ( L, R, N ) , ℓ -pregroups where N ⊆ L × R . This is an algebraic rendering of sequents! Embedding theorems Maps on a chain Nick Galatos, BLAST, August 2013 Developments on higher levels of the substructural hierarchy – 3 / 20
Lattice frames Residuated lattices Lattice frames Formula hierarchy Residuated frames GN FL Frame applications c a c d d ≤ a d c Hypersequents Frame constructions a × × Hyper-frames a a c b b Examples × × b Extra structure Hyper and PUFs × c Extensions Diagrams ALG and MV We obtain an oriented bipartite graph/ lattice frame F = ( L, R, N ) , ℓ -pregroups where N ⊆ L × R . This is an algebraic rendering of sequents! Embedding theorems Maps on a chain Given X ⊆ L and Y ⊆ R we define X ⊲ = { b ∈ R : x N b, for all x ∈ X } Y ⊳ = { a ∈ L : a N y, for all y ∈ Y } Then γ N ( X ) = X ⊲⊳ defines a closure operator on P ( L ) and the Galois algebra F + = ( γ N [ P ( L )] , ∩ , ∪ γ N ) is a complete lattice. Nick Galatos, BLAST, August 2013 Developments on higher levels of the substructural hierarchy – 3 / 20
Lattice frames Residuated lattices Lattice frames Formula hierarchy Residuated frames GN FL Frame applications c a c d d ≤ a d c Hypersequents Frame constructions a × × Hyper-frames a a c b b Examples × × b Extra structure Hyper and PUFs × c Extensions Diagrams ALG and MV We obtain an oriented bipartite graph/ lattice frame F = ( L, R, N ) , ℓ -pregroups where N ⊆ L × R . This is an algebraic rendering of sequents! Embedding theorems Maps on a chain Given X ⊆ L and Y ⊆ R we define X ⊲ = { b ∈ R : x N b, for all x ∈ X } Y ⊳ = { a ∈ L : a N y, for all y ∈ Y } Then γ N ( X ) = X ⊲⊳ defines a closure operator on P ( L ) and the Galois algebra F + = ( γ N [ P ( L )] , ∩ , ∪ γ N ) is a complete lattice. If A is a lattice, F A = ( A, A, ≤ ) is a lattice frame. Also, F + A is the Dedekind-MacNeille completion of A and x �→ { x } ⊳ is an embedding. Nick Galatos, BLAST, August 2013 Developments on higher levels of the substructural hierarchy – 3 / 20
Formula hierarchy Residuated lattices Lattice frames Formula hierarchy Residuated frames GN ♣♣♣♣♣♣♣♣♣ ✻ ♣♣♣♣♣♣♣♣♣ ✻ FL Frame applications Hypersequents We separate our signature into positive ■ Frame constructions Hyper-frames {∨ , · , 1 } and negative {∧ , \ , / } . P 3 N 3 Examples The sets P n , N n of formulas are defined by: ■ Extra structure ✻ ❅ ■ � ✒ ✻ ❅ � Hyper and PUFs P 0 = N 0 = the set of variables (0) � ❅ Extensions P n +1 = �N n � � , � Diagrams (P) � ❅ ALG and MV N n +1 = �P n � � , P n +1 \ ,/ P n +1 (N) ℓ -pregroups P 2 N 2 Embedding theorems P 1 -reduced: � � p i ✻ ■ ❅ ❅ � ✒ ✻ � Maps on a chain ■ � ❅ N 1 -reduced: � ( p 1 p 2 · · · p n \ r/q 1 q 2 · · · q m ) ■ � ❅ p 1 p 2 · · · p n q 1 q 2 · · · q m ≤ r P 1 N 1 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. Nick Galatos, BLAST, August 2013 Developments on higher levels of the substructural hierarchy – 4 / 20
Residuated frames Residuated lattices Lattice frames Formula hierarchy Residuated frames A residuated frame is a structure F = ( L, R, N, ◦ , ε, � , � ) , where L GN FL and R are sets N ⊆ L × R , Frame applications Hypersequents Frame constructions Hyper-frames Examples Extra structure Hyper and PUFs Extensions Diagrams ALG and MV ℓ -pregroups Embedding theorems Maps on a chain Nick Galatos, BLAST, August 2013 Developments on higher levels of the substructural hierarchy – 5 / 20
Residuated frames Residuated lattices Lattice frames Formula hierarchy Residuated frames A residuated frame is a structure F = ( L, R, N, ◦ , ε, � , � ) , where L GN FL and R are sets N ⊆ L × R , Frame applications Hypersequents L = ( L, ◦ , ε ) is a monoid ■ Frame constructions Hyper-frames Examples Extra structure Hyper and PUFs Extensions Diagrams ALG and MV ℓ -pregroups Embedding theorems Maps on a chain Nick Galatos, BLAST, August 2013 Developments on higher levels of the substructural hierarchy – 5 / 20
Residuated frames Residuated lattices Lattice frames Formula hierarchy Residuated frames A residuated frame is a structure F = ( L, R, N, ◦ , ε, � , � ) , where L GN FL and R are sets N ⊆ L × R , Frame applications Hypersequents L = ( L, ◦ , ε ) is a monoid ■ Frame constructions R is an L -biset under � : L × R → R and � : R × L → R ■ Hyper-frames Examples Extra structure Hyper and PUFs Extensions Diagrams ALG and MV ℓ -pregroups Embedding theorems Maps on a chain Nick Galatos, BLAST, August 2013 Developments on higher levels of the substructural hierarchy – 5 / 20
Residuated frames Residuated lattices Lattice frames Formula hierarchy Residuated frames A residuated frame is a structure F = ( L, R, N, ◦ , ε, � , � ) , where L GN FL and R are sets N ⊆ L × R , Frame applications Hypersequents L = ( L, ◦ , ε ) is a monoid ■ Frame constructions R is an L -biset under � : L × R → R and � : R × L → R ■ Hyper-frames Examples ( x ◦ y ) N z ⇔ y N ( x � z ) ⇔ x N ( z � y ) ■ Extra structure Hyper and PUFs Extensions Diagrams ALG and MV ℓ -pregroups Embedding theorems Maps on a chain Nick Galatos, BLAST, August 2013 Developments on higher levels of the substructural hierarchy – 5 / 20
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