Epimorphisms in varieties of square-increasing residuated structures - PowerPoint PPT Presentation

Epimorphisms in varieties of square-increasing residuated structures T. Moraschini 1 , J.G. Raftery 2 , and J.J. Wannenburg 2 , 3 1 Academy of Sciences of the Czech Republic, Czech Republic 2 University of Pretoria, South Africa 3 DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS) LATD, August 2018 Opinions expressed and conclusions arrived at are those of the author and are not necessarily to be attributed to the CoE-MaSS.

ES Property ◮ Let K be a variety of algebras. Let ❆ , ❇ ∈ K and f : ❆ → ❇ a homomorphism. ◮ f is a K -epimorphism if, whenever ❈ ∈ K and g , h : ❇ → ❈ are homomorphisms such that g ◦ f = h ◦ f , then f = g . ◮ Note that if f is surjective, then f is a K -epimorphism. ◮ If all K -epimorphisms are surjective, then K has the epimorphism surjectivity (ES) property . ◮ ❆ is called a K -epic subalgebra of ❇ if the injection map i : ❆ → ❇ is a K -epimorphism. ◮ K has surjective epimorphisms iff there is no ❇ ∈ K with a proper K -epic subalgebra. ✓✏ ✓✏ ✓✏ g ✲ f ✲ ✲ ❆ ❇ ❈ ✒✑ ✒✑ ✒✑ h

Beth property If a variety K algebraizes a logic ⊢ , then K has the ES property iff ⊢ has the infinite (deductive) Beth (definability) property, i.e., whenever a set Z of variables is defined implicitly in terms of a disjoint set X of variables by means of some set Γ of formal assertions about X ∪ Z , then Γ also defines Z explicitly in terms of X .

Residuated structures A square-increasing commutative residuated lattice (SRL) ❆ = � A ; ∧ , ∨ , · , → , e � comprises ◮ a lattice � A ; ∧ , ∨� , ◮ a commutative monoid � A ; · , e � ◮ that is square-increasing, i.e., x ≤ x · x := x 2 , ◮ and a binary operation → satisfying the law of residuation x · y ≤ z iff y ≤ x → z . We may enrich the language of SRLs with a constant symbol f and define an involution ¬ x := x → f that satisfies x = ¬¬ x and x → ¬ y = y → ¬ x , thus obtaining SIRLs. It follows that ¬ e = f .

Residuated structures ◮ In any SRL if x , y ≤ e then x · y = x ∧ y . ◮ Integral ( x ≤ e ) SRLs are called Brouwerian algebras in which case · coincides with ∧ . Given an S[I]RL ❆ , its negative cone A − := { a ∈ A : a ≤ e } can be turned into a Brouwerian algebra ❆ − = � A − ; ∧ , ∨ , e , → − � , by restricting the operations ∧ , ∨ to A − and defining a → − b = ( a → b ) ∧ e , for a , b ∈ A − . ◮ Heyting algebras are Brouwerian algebras with a distinguished bottom element ⊥ . ◮ Distributive SRLs are called Dunn monoids . ◮ Distributive SIRLs are called De Morgan monoids . ◮ Idempotent ( x = x · x ) De Morgan monoids are called Sugihara monoids . Their structure is very well understood.

Logic The varieties of algebras on the left algebraize the logics on the right. LR t + SRLs LR t SIRLs R t + Dunn monoids R t De Morgan monoids RM t Sugihara monoids Brouwerian Algebras positive intuitionistic logic Heyting Algebras intuitionistic logic

Depth For any Brouwerian algebra ❇ , let Pr( ❇ ) be the set of non-empty prime lattice filters F of ❇ , including B itself. We define the depth of F in Pr( ❇ ) to be the greatest n ∈ ω (if it exists) such that there is a chain in Pr( ❇ ) of the the form F = F 0 � F 1 � · · · � F n = B . If no greatest such n exists, we say F has depth ∞ in Pr( ❇ ). We define depth( ❇ ) = sup { depth( F ) : F ∈ Pr( ❇ ) } . If ❆ is an S[I]RL and K is a variety of S[I]RLs, we define depth( ❆ ) = depth( ❆ − ) and depth( K ) = sup { depth( ❆ ) : ❆ ∈ K } . If a variety of S[I]RLs is finitely generated, then it has finite depth (but not conversely).

Main Theorem Thm. Let K be a variety of SRLs such that 1) ❆ = Sg( A − ) for every ❆ ∈ K FSI and 2) K has finite depth. Then K has surjective epimorphisms. ◮ K FSI denotes the class of finitely subdirectly irreducible members of K . Sg( − ) denotes subalgebra generation. ◮ The argument also works for expansions of SRLs, where the additional operations on each algebra in K are compatible with the congruences of its SRL reduct. ◮ In particular, the theorem holds if we add involution or distinguished least elements to the type. ◮ Therefore, this theorem generalises the recent result of Bezhanishvili, Moraschini and Raftery (2017) that every variety of Brouwerian or Heyting algebras of finite depth has the ES property.

(1) can’t be dropped Let K be a variety of S[I]RLs such that 1) ❆ = Sg( A − ) for every ❆ ∈ K FSI and Thm. 2) K has finite depth. Then K has surjective epimorphisms. Urquhart (1999) showed that the crystal lattice has a proper epic ( f ) 2 s subalgebra in the variety of De crystal: f s Morgan monoids. � ❅ Its involution-less reduct also has s � ❅ s a = ¬ a b = ¬ b ❅ s � a proper epic subalgebra in the ❅ � variety of Dunn monoids. e s ¬ ( f 2 )

(2) can’t be dropped for SRLs Let K be a variety of S[I]RLs such that 1) ❆ = Sg( A − ) for every ❆ ∈ K FSI and Thm. 2) K has finite depth. Then K has surjective epimorphisms. s Bezhanishvili, Moraschini and e s Raftery (2017) exhibit a � ❅ ❆ 1 : s � ❅ s Brouwerian algebra of infinite b 0 c 0 ❅ s � depth that generates a variety ❅ � s � ❅ without the ES property. s � ❅ s b 1 c 1 ❅ s � ❅ � � ❅ q q q

Reflection s s e D s s a Dunn monoid

Reflection s s e s D s a Dunn monoid

Reflection s s s e D s a Dunn monoid

Reflection s s s e D s a Dunn monoid

Reflection s s s e D s a Dunn monoid

Reflection s s s e D s a Dunn monoid

Reflection s s s e D s a Dunn monoid

Reflection s s s e D s a Dunn monoid

Reflection s ⊤ s a ′ D ′ s e ′ R ( D ) s e D s a ⊥ s

Reflection ◮ There is a unique way of turning the structure into a De Morgan monoid R ( ❉ ) = � D ∪ D ′ ∪ {⊥ , ⊤} ; ∧ , ∨ , · , ¬ , e � of which ❉ is a subreduct and where ¬ extends ′ . ◮ If K is a class of Dunn monoids we define R ( K ) := { R ( ❉ ) : ❉ ∈ K } . Thm. A Dunn monoid ❉ is generated by its negative cone iff R ( ❉ ) is. Let K be a variety of Dunn monoids. Then K has Thm. surjective epimorphisms iff V ( R ( K )) has. In particular, ❉ has a proper K -epic subalgebra iff R ( ❉ ) has a proper V ( R ( K )) -epic subalgebra.

(2) can’t be dropped for SIRLs Let K be a variety of S[I]RLs such that 1) ❆ = Sg( A − ) for every ❆ ∈ K FSI and Thm. 2) K has finite depth. Then K has surjective epimorphisms. s ⊤ q q q � ❅ s � ❅ � ❅ s s � ❅ � c ′ ❅ b ′ s 0 0 One can show that R ( ❆ 1 ) has s e ′ infinite depth and has a proper R ( ❆ 1 ) : s V ( R ( ❆ 1 ))-epic subalgebra. e s � ❅ � s ❅ s b 0 c 0 ❅ s � ❅ � � ❅ q q q s ⊥

Applications Let K be a variety of S[I]RLs such that 1) ❆ = Sg( A − ) for every ❆ ∈ K FSI and Thm. 2) K has finite depth. Then K has surjective epimorphisms. There are many interesting varieties of S[I]RLs whose FSI members are generated by their negative cones. For example Sugihara monoids, their involution-less subreducts, and Brouwerian algebras. Our main theorem therefore implies (the known results) that each of their finitely generated subvarieties have surjective epimorphisms.

New applications Let K be a variety of S[I]RLs such that 1) ❆ = Sg( A − ) for every ❆ ∈ K FSI and Thm. 2) K has finite depth. Then K has surjective epimorphisms. ◮ ❈ 4 is the only 0-generated algebra onto which finitely subdirectly irreducible De Morgan s f 2 monoids may be mapped by non-injective s f homomorphisms. ❈ 4 : s e ◮ There is a largest variety U of De Morgan s ¬ ( f 2 ) monoids consisting of homomorphic preimages of ❈ 4 (and trivial algebras). DMM U V ( ❈ 4 ) trivial

New applications Let K be a variety of S[I]RLs such that 1) ❆ = Sg( A − ) for every ❆ ∈ K FSI and Thm. 2) K has finite depth. Then K has surjective epimorphisms. V ( ❈ 4 ) has just ten covers in U , each of which satisfies (1) and is generated by a finite algebra, see (Moraschini, Raftery and Wannenburg, 2017). Each of these varieties therefore has surjective epimorphisms. DM U V ( ❈ 4 ) trivial

Semilinear SRLs An SRL ❆ is called semilinear if it is a subdirect product of totally ordered SRLs. These form a variety. A semilinear S[I]RL is FSI iff it is totally ordered. Certain semilinear varieties of S[I]RLs enjoy strong positive ES results, see (Bezhanishvili, Moraschini and Raftery, 2017): ◮ Every variety of semilinear Brouwerian or Heyting algebras (resp. relative Stone and G¨ odel algebras) has surjective epimorphisms. ◮ Every variety of Sugihara monoids is semilinear and has surjective epimorphisms, ◮ the same applies to the involution-less subreducts of Sugihara monoids.