A Categorial Equivalence for semi-Nelson algebras
aa r X i v : . [ m a t h . L O ] J u l A Categorial Equivalence for semi-Nelsonalgebras
Juan Manuel Cornejo , Andr´es Gallardo , and IgnacioViglizzo INMABB UNS CONICET, Departamento de Matem´atica -Av. Alem 1253, Baha Blanca, Buenos Aires, Argentina. * Corresponding author: [email protected] ,ORCiD:0000-0002-5303-623XJuly 30, 2020
Abstract
We present a category equivalent to that of semi-Nelson algebras.The objects in this category are pairs consisting of a semi-Heytingalgebra and one of its filters. The filters must contain all the denseelements of the semi-Heyting algebra and satisfy an additional tech-nical condition. We also show that in the case of dually hemimorphicsemi-Nelson algebras, the filters are not necessary and the category isequivalent to that of dually hemimorphic semi-Heyting algebras.
Semi-Heyting algebras were introduced in 1985 by H. Sankappanavar [San85]as a variety generalizing that of Heyting algebras while keeping many of itsgood features like being distributive pseudocomplemented lattices and havingtheir congruences determined by filters.On the other hand, D. Vakarelov provided in [Vak77] a way of construct-ing Nelson algebras from Heyting ones, by means of what is now known as1wist product, thus extending work by Kalman [Kal58]. This program wascontinued by Sendlewski [Sen84], [Sen90], who gave a full representation ofNelson algebras using Heyting algebras and a boolean congruence on them.Later, L. Monteiro and I. Viglizzo [Vig99] and [MV19], used filters insteadof congruences to study this representation.It turned out that another one of the nice features of semi-Heyting alge-bras is that Vakarelov’s construction can be carried out on them as well.The resulting algebras form the variety of semi-Nelson algebras [CV16].Vakarelov’s construction gives a categorial equivalence between semi-Heytingalgebras and centered semi-Nelson algebras [CSM18], but this leaves outmany semi-Nelson algebras. In this work we prove that considering pairs ofsemi-Heyting algebras and one of its filters satisfying some extra conditions,we can construct all the semi-Nelson algebras. This gives a good representa-tion that can be seen as a categorial equivalence, and permits understandingsemi-Nelson algebras in terms of the better-known semi-Heyting algebras.H. Sankappanavar defined in [San11] the variety of semi-Heyting algebraswith a dual hemimorphism, or dually hemimorphic semi-Heyting algebras,
DSH , as an expansion of semi-Heyting algebras with an unary operator thatis a common generalization of both De Morgan and pseudocomplementedalgebras. J. M. Cornejo and H. San Mart´ın applied Vakarelov’s constructionto this expansion to obtain the variety of dually hemimorphic semi-Nelson al-gebras,
DSN [CSM18]. We prove here that dually hemimorphic semi-Nelsonalgebras are centered, and therefore they can be represented by dually hemi-morphic semi-Heyting algebras.The paper is organized as follows: Section 2 introduces the varieties ofalgebras we will be dealing with, recalls some of the basic results on themthat we will be using, and provides background on Vakarelov’s construction.Section 3 details the representation of semi-Nelson algebras by pairs con-sisting of a semi-Heyting algebra and one of its filters containing the denseelements of the algebra and satisfying an additional condition, proving thisrepresentation is an equivalence of two categories. The final section dealswith the case of dually hemimorphic semi-Nelson algebras.2
Preliminaries
In this section we will recall the definitions and some basic properties of thevarieties of Heyting and semi-Heyting algebras, Nelson and semi-Nelson al-gebras, pseudocomplemented lattices, and dually hemimorphic semi-Heytingand semi-Nelson algebras.
Definition 2.1 A semi-Heyting algebra is an algebra H = h H ; ∧ , ∨ , ⇒ , , i of type (2 , , , , such that the following conditions are satisfied for all x, y, z ∈ A :(SH1) h H, ∧ , ∨ , , i is a bounded lattice,(SH2) x ∧ ( x ⇒ y ) = x ∧ y ,(SH3) x ∧ ( y ⇒ z ) = x ∧ (( x ∧ y ) ⇒ ( x ∧ z )) ,(SH4) x ⇒ x = 1 .A semi-Heyting algebra is a Heyting algebra if it satisfies the identity:(H) ( x ∧ y ) ⇒ x = 1 . On a semi-Heyting algebra H one can always define the term x ⇒ H y = x ⇒ ( x ∧ y ). With this operation, the system h H, ∧ , ∨ , ⇒ H , , i is a Heytingalgebra [ACDV13].Nelson algebras are the algebraic counterpart of D. Nelson’s constructivelogic with strong negation. The equations presented here were proved to bea complete and independent axiomatization in [MM96]: Definition 2.2 A Nelson algebra is an algebra A = h A ; ∧ , ∨ , → , ∼ , i oftype (2 , , , , such that the following conditions are satisfied for all x, y, z ∈ A :(N1) x ∧ ( x ∨ y ) = x ,(N2) x ∧ ( y ∨ z ) = ( z ∧ x ) ∨ ( y ∧ x ) ,(N3) ∼∼ x = x , N4) ∼ ( x ∧ y ) = ∼ x ∨ ∼ y ,(N5) x ∧ ∼ x = ( x ∧ ∼ x ) ∧ ( y ∨ ∼ y ) ,(N6) x → x = 1 ,(N7) x → ( y → z ) = ( x ∧ y ) → z ,(N8) x ∧ ( x → y ) = x ∧ ( ∼ x ∨ y ) . The variety of semi-Nelson algebras was introduced in [CV16] as a gen-eralization of Nelson algebras.
Definition 2.3 A semi-Nelson algebra is an algebra A = h A ; ∧ , ∨ , → , ∼ , i of type (2 , , , , such that the following conditions are satisfied for all x, y, z ∈ A :(SN1) x ∧ ( x ∨ y ) = x ,(SN2) x ∧ ( y ∨ z ) = ( z ∧ x ) ∨ ( y ∧ x ) ,(SN3) ∼∼ x = x ,(SN4) ∼ ( x ∧ y ) = ∼ x ∨ ∼ y ,(SN5) x ∧ ∼ x = ( x ∧ ∼ x ) ∨ ( y ∨ ∼ y ) ,(SN6) x ∧ ( x → N y ) = x ∧ ( ∼ x ∨ y ) ,(SN7) x → N ( y → N z ) = ( x ∧ y ) → N z ,(SN8) ( x → N y ) → N [( y → N x ) → N [( x → z ) → N ( y → z )]] = 1 ,(SN9) ( x → N y ) → N [( y → N x ) → N [( z → x ) → N ( z → y )]] = 1 ,(SN10) ( ∼ ( x → y )) → N ( x ∧ ∼ y ) = 1 ,(SN11) ( x ∧ ∼ y ) → N ( ∼ ( x → y )) = 1 ,where x → N y := x → ( x ∧ y ) . A , the system h A, ∧ , ∨ , → N , ∼ , i is a Nelsonalgebra [CV16].We will denote by H , SH , N and SN the varieties of Heyting, semi-Heyting, Nelson and semi-Nelson algebras respectively.Heyting algebras, as is well known, are pseudocomplemented (with x ∗ = x → x ∈ H ) in the sense of the followingdefinition: Definition 2.4
An algebra A = h A ; ∧ , ∨ , ∗ , , i is a pseudocomplementedlattice if the following conditions hold:PS1) h A ; ∧ , ∨ , , i is a bounded lattice,PS2) x ∧ ( x ∧ y ) ∗ = x ∧ y ∗ ,PS3) ∗ = 1 and ∗ = 0 . If we define in a semi-Heyting algebra the term x c = x ⇒
0, it turns outthat x c is the pseudocomplement of x in the sense of the previous definition.Moreover, x ⇒ H x ⇒ ( x ∧
0) = x ⇒
0, so x ∗ and x c coincide in SH .From now on we will use indistinctly the notation x ∗ for both x c and x ∗ .We will use some properties of pseudocomplemented lattices. Their proofcan be found in [BD74]: Proposition 2.5 If A = h A ; ∧ , ∨ , ∗ , , i is a pseudocomplemented lattice,then for every x, y ∈ A , if x ∧ y = 0 , then x ≤ y ∗ . It also holds that ( x ∨ y ) ∗ = x ∗ ∧ y ∗ . Definition 2.6
Let A = h A ; ∧ , ∨ , ∗ , , i be a pseudocomplemented lattice.We say that an element a ∈ A is dense if a ∗ = 0 . We denote by D s ( A ) theset of all dense elements of A . The following characterization of dense elements is going to be useful:
Lemma 2.7 If H ∈ SH , then x ∈ D s ( H ) if and only if x = y ∨ y ∗ for some y ∈ H . For the results in section 4, we now define the varieties of dually hemimor-phic semi-Heyting and semi-Nelson algebras. The former were introduced byH.P. Sankappanavar in [San11], and the latter were presented in [CSM19].5 efinition 2.8
An algebra A = h A ; ∧ , ∨ , → , † , , i of type (2 , , , , , issaid to be a dually hemimorphic semi-Heyting algebra if h A ; ∧ , ∨ , → , , i isa semi-Heyting algebra and the following equations are satisfied:DSM1) † = 1 ,DSM2) † = 0 ,DSM3) ( x ∧ y ) † = x † ∨ y † . We write
DSH to denote the variety of dually hemimorphic semi-Heytingalgebras.
Definition 2.9
An algebra A = h A ; ∧ , ∨ , → , ∼ , ′ , i of type (2 , , , , issaid to be a dually hemimorphic semi-Nelson algebra if h A ; ∧ , ∨ , → , ∼ , i isa semi-Nelson algebra and the following equations are satisfied:DSN1) ( ∼ ′ = 1 ,DSN2) ′ → ( ∼
1) = 1 ,DSN3) (( x → y ) ∧ ( y → x ) ∧ x ′ ) → (( x → y ) ∧ ( y → x ) ∧ y ) ′ = 1 ,DSN4) ∼ x ′ → ( ∼ x ∧ ( x ′ → x )) = 1 ,DSN5) ( ∼ x ∧ ( x ′ → x )) →∼ x ′ = 1 ,DSN6) ( x ∧ y ) ′ → ( x ′ ∨ y ′ ) = 1 ,DSN7) ( x ′ ∨ y ′ ) → ( x ∧ y ) ′ = 1 .We use the convention that the unary operation ′ has higher priority than ∼ , so the expression ∼ x ′ means ∼ ( x ′ ) . We write
DSN to denote the variety of dually hemimorphic semi-Nelsonalgebras. 6 .2 Vakarelov’s Construction
In this section we review some well known results that established the connec-tion between Heyting and Nelson algebras, and later allowed the definitionof (dually hemimorphic) semi-Nelson algebras as a variety constructed fromthe one of (dually hemimorphic) semi-Heyting algebras.Let A ∈ H . We denote with V k ( A ) the set { ( a, b ) ∈ A : a ∧ b = 0 } andwith V k ( A ) the system h V k ( A ); ⊓ , ⊔ , → , ∼ , ⊤i algebrized in the following way(V1) ( a, b ) ⊓ ( c, d ) = ( a ∧ c, b ∨ d ),(V2) ( a, b ) ⊔ ( c, d ) = ( a ∨ c, b ∧ d ),(V3) ( a, b ) → ( c, d ) = ( a ⇒ c, a ∧ d ),(V4) ∼ ( a, b ) = ( b, a ),(V5) ⊤ = (1 , Observation 2.10
From the previous definitions we can deduce the rule (V6) ( a, b ) → N ( c, d ) = ( a ⇒ H c, a ∧ d ) . It was shown in [Vak77] that V k ( A ) ∈ SN .Vakarelov’s construction can be generalized the following way: if A ∈ H ,and F is a filter of A , we define the structure N ( A, F ) := { ( a, b ) ∈ A : a ∧ b = 0 , a ∨ b ∈ F } . It was proved in [Vig99, MV19] that N ( A, F ), algebrized with (V1)-(V5),is well defined and it is a Nelson algebra. This algebra will be denoted by N ( A , F ).A different generalization was to start from a semi-Heyting algebra, whichlead to the definition of semi-Nelson algebras [CV16]: if A is a semi-Heytingalgebra, then V k ( A ) is a semi-Nelson algebra.Going in the other direction, we can go from semi-Nelson algebras tosemi-Heyting algebras by means of the following quotient:Let A = h A, ∧ , ∨ , → , ∼ , i ∈ SN , and define: x ≡ y if and only if x → y = 1 and y → x = 1 . It can be proved that “ ≡ ” is an equivalence relation and a congruence withrespect of the operations ∧ , ∨ and → . We denote with [[ a ]] the equivalenceclass of an element a ∈ A under the relation ≡ .We consider sH ( A ) = h A/ ≡ , ∩ , ∪ , ⇒ , ⊥ , ⊤i , where7 ⊥ = [[ ∼ • ⊤ = [[1]], • [[ x ]] ∩ [[ y ]] = [[ x ∧ y ]], • [[ x ]] ∪ [[ y ]] = [[ x ∨ y ]], • [[ x ]] ⇒ [[ y ]] = [[ x → y ]].Then sH ( A ) ∈ SH ([CV16]).We will use the next result in the following sections: Proposition 2.11 [CV16, Lemma 2.7 (h) and (l)]
Let A ∈ SN , and a, b, c ∈ A . Then the following properties hold:(1) a → N b = b → N a = 1 if and only if a → b = b → a = 1 ,(2) If a → N b = 1 = b → N c = 1 , then a → N c = 1 . Part (1) of the previous proposition tells us that the congruence ≡ is thesame as the one that can be defined if we replace → with → N .The following representation theorem establishes the connection betweensemi-Heyting and semi-Nelson algebras using Vakarelov’s construction andthe quotient by the relation ≡ . Theorem 2.12 ([CV16, Theorem 5.3 and Corollary 5.2]) If A ∈ SH , then A is isomorphic to sH ( V k ( A )) through the isomorphism g ( a ) = [[( a, a ∗ )]] .Also, if A ∈ SN , then A is isomorphic to a subalgebra of V k ( sH ( A )) throughthe application h ( a ) = ([[ a ]] , [[ ∼ a ]]) . Theorem 2.12 gives a representation of any semi-Nelson algebra A as a sub-algebra of V k ( sH ( A )), but there can be more than one non-isomorphic semi-Nelson algebra having the same quotient semi-Heyting algebra sH ( A ) (upto isomorphism). In order to get a sharper representation we are going toconsider pairs consisting of a semi-Heyting algebra and one of its filters satis-fying extra conditions. To be more precise, we obtain a categorial equivalencebetween the category of semi-Nelson algebras and a category whose objectsare pairs ( H , F ) where H is a semi-Heyting algebra and F is a filter of H that contains all the dense elements of H and satisfies an extra condition.8ecall the construction N ( A, F ) = { ( a, b ) ∈ A : a ∧ b = 0 , a ∨ b ∈ F } forsome Heyting algebra A , and F a filter of A . For any A ∈ H , we consider V k ( A ), and a subalgebra S such that the projection over the first component, π , verifies π ( S ) = A . It was shown in [Vig99] that there is a filter F ⊆ A containing the dense elements of A such that S = N ( A , F ).To extend these results to the variety SH , we define: Definition 3.1
Let H ∈ SH . A subset F ⊆ H is an i -filter, if the followingconditions hold:IF1) F is a lattice filter of H .IF2) D s ( H ) ⊆ F .IF3) If z ∨ t ∈ F , then for all x ∈ H , ( x ⇒ z ) ∨ ( x ∧ t ) ∈ F . It is immediate from the definition that every i -filter is a filter. Theconverse is not true, as the following example shows: Example 3.2
Consider the semi-Heyting algebra A = h A ; ∧ , ∨ , ⇒ , i , withthe operation ⇒ indicated in the table: ❡❡ A ⇒ The subset F = { } of A is a filter, and it satisfies IF (since the onlydense element is 1). But F does not satisfy IF ) because ∨ ∈ F , but (0 ⇒ ∨ (0 ∧
0) = 0 ∨ / ∈ F . Observation 3.3
If we consider the structure N ( H , F ) , with H ∈ SH , an i -filter F , and ( a, b ) , ( c, d ) ∈ N ( H, F ) , then (1) c ∧ d = 0 and (2) c ∨ d ∈ F .Notice that ( a ⇒ c ) ∧ ( a ∧ d ) = a ∧ ( a ⇒ c ) ∧ d = ( SH ) ( a ∧ c ) ∧ d = a ∧ ( c ∧ d ) = (1) a ∧ . Also, using (2) and IF we have that ( a ⇒ c ) ∨ ( a ∧ d ) ∈ F , so the pair ( a ⇒ c, a ∧ d ) is in N ( H, F ) and we can use it to define ( a, b ) → ( c, d ) =( a ⇒ c, a ∧ d ) . emma 3.4 If H ∈ SH and F is an i -filter of H , then the system h N ( H, F ); ⊓ , ⊔ , → , ∼ , ⊤i with the operations defined as in (V1)-(V5) is asemi-Nelson algebra. Proof.
Using that F is a filter we can prove that the operations ⊓ , ⊔ , ⊤ and ∼ are well defined in N ( H, F ) (see [MV19]). Also, by observation 3.3,it follows that the operation → is well defined. Furthermore, following theproof of Theorem 4.1 in[CV16], N ( H , F ) verifies the axioms SN SN
11) soit is is a semi-Nelson algebra. (cid:4)
Lemma 3.5
Let H ∈ SH . If S is a subalgebra of V k ( H ) , such that π ( S ) = H (with π the projection over the first component of V k ( H ) ), then thereexists an i -filter E of H such that S = N ( H, E ) . Proof.
Let E = { x ∈ H : x = a ∨ b for some ( a, b ) ∈ S } . We are going toprove that E is an i -filter.It was shown in [Vig99] that if H ∈ H , then E is a filter of H . Butthat proof can be extended naturally to the variety SH , because it only useslattice properties.Consider a dense element x ∈ H . Since π ( S ) = H , then there exists y ∈ H such that ( x, y ) ∈ S . Besides, since S is a subalgebra of V k ( H ), (0 , ∈ S ,and therefore ( x, y ) → (0 ,
1) = ( x ⇒ , x ∧
1) = ( x ∗ , x ) = (0 , x ) ∈ S . Hence, x ∈ E because x = x ∨
0, and we conclude that E satisfies IF z, t ∈ H we have z ∨ t ∈ E ,Since z ∨ t ∈ E , there exists a pair ( a, b ) ∈ S such that z ∨ t = a ∨ b .But S is a subalgebra of V k ( H ), so ∼ ( a, b ) = ( b, a ) ∈ S , and therefore( a, b ) ∪ ( b, a ) = ( a ∨ b, a ∧ b ) = ( a ∨ b, ∈ S . This means that ( z ∨ t, ∈ S ,and hence ∼ ( z ∨ t,
0) = (0 , z ∨ t ) ∈ S. (3.1)Using the hypothesis π ( S ) = H , and x, z, t ∈ H , it follows that thereexist some x ′ , z ′ , t ′ ∈ H such that ( x, x ′ ) , ( z, z ′ ) , ( t, t ′ ) ∈ S . This implies thatthe next term belongs to S :(( x, x ′ ) → ( z, z ′ )) ∪ (( x, x ′ ) ∩ ( t, t ′ )) = ( x ⇒ z, x ∧ z ′ ) ∪ ( x ∧ t, x ′ ∨ t ′ ) =(( x ⇒ z ) ∨ ( x ∧ t ) , ( x ∧ z ′ ) ∧ ( x ′ ∨ t ′ )) . (3.2)Since ( x ∧ z ′ ) ∧ ( x ′ ∨ t ′ ) = ( x ∧ z ′ ∧ x ′ ) ∨ ( x ∧ z ′ ∧ t ′ ) = 0 ∨ ( x ∧ z ′ ∧ t ′ ) = x ∧ z ′ ∧ t ′ ,(3.2) yields (( x ⇒ z ) ∨ ( x ∧ t ) , x ∧ z ′ ∧ t ′ ) ∈ S. (3.3)10ombining (3.1) with (3.3) we get(( x ⇒ z ) ∨ ( x ∧ t ) , x ∧ z ′ ∧ t ′ ) ∪ (0 , z ∨ t ) = (( x ⇒ z ) ∨ ( x ∧ t ) , x ∧ z ′ ∧ t ′ ∧ ( z ∨ t )) ∈ S. (3.4)Since x ∧ z ′ ∧ t ′ ∧ ( z ∨ t ) = ( x ∧ z ′ ∧ t ′ ∧ z ) ∧ ( x ∧ z ′ ∧ t ′ ∧ t ) = ( x ∧ t ′ ∧ ( z ∧ z ′ )) ∨ ( x ∧ z ∧ ( t ∧ t ′ )) = 0 ∨ x ⇒ z ) ∨ ( x ∧ t ) , ∈ S. By the definition of E it follows that( x ⇒ z ) ∨ ( x ∧ t ) ∨ x ⇒ z ) ∨ ( x ∧ t ) ∈ E. Hence, E is an i -filter.Let us prove now that N ( H, E ) = S . It is clear that S ⊆ N ( H, E ). Ifwe have ( a, b ) ∈ N ( H, E ), then a ∧ b = 0, and a ∨ b ∈ E , so there exists( c, d ) ∈ S such that c ∨ d = a ∨ b . Then, as before ( c, d ) ∪ ( d, c ) = ( c ∨ d,
0) =( a ∨ b, ∈ S . On the other hand, since π ( S ) = H , there exists some b ′ ∈ H such that ( b, b ′ ) ∈ S , and therefore ( b, b ′ ) → (0 ,
1) = ( b ∗ , b ) ∈ S .By properties of the pseudocomplement, a ∧ b = 0 implies that a ≤ b ∗ andhence ( a ∨ b ) ∧ b ∗ = ( a ∧ b ∗ ) ∨ ( b ∧ b ∗ ) = a ∨ a , so we can conclude that( a ∨ b, ∩ ( b ∗ , b ) = (( a ∨ b ) ∧ b ∗ , ∨ b ) = ( a, b ) ∈ S . (cid:4) Theorem 3.6
Every semi-Nelson algebra A is isomorphic to an algebra ofthe form N ( H , F ) , for some semi-Heyting algebra H and some i -filter F of H . Proof.
By Theorem 2.12, every semi-Nelson algebra A can be obtained asa subalgebra of V k ( sH ( A )), and clearly π ( V k ( sH ( A ))) = sH ( A ), so A isisomorphic to N ( sH ( A ) , E ), where E is the i -filter of Lemma 3.5. (cid:4) Much in the same manner as for the case of Heyting algebras, for H ∈ SH ,the i -filters E such that π ( N ( H, E )) = H are exactly the ones that containthe dense elements of H : Lemma 3.7 If H ∈ SH , and F is an i -filter of H , then π ( N ( H, F )) = H . Proof. If x ∈ H , by Lemma 2.7, we have that x ∨ x ∗ ∈ D s ( H ) ⊆ F . Also, x ∧ x ∗ = 0 is valid, so ( x, x ∗ ) ∈ N ( H, F ) and x ∈ π ( N ( H, F )). (cid:4) .1 Categorial equivalence We define now the category sHF , which has as objects pairs ( H , F ) (where H ∈ SH , and F is an i -filter of H ), and as morphisms, functions f :( H , F ) −→ ( H ′ , F ′ ) such that f : H −→ H ′ is a homomorphism of semi-Heyting algebras and f ( F ) ⊆ F ′ .Consider now the category sN of semi-Nelson algebras and their mor-phisms; our objective now is to establish an equivalence between these cate-gories. For this we define functors α : sHF −→ sN and β : sN −→ sHF . Proposition 3.8
The application α : sHF −→ sN defined on the objects of sHF by α (( H , F )) = N ( H , F ) , and on morphisms by α ( f )( a, b ) = ( f ( a ) , f ( b )) for all ( a, b ) ∈ N ( H, F ) , is a functor from sHF to sN . Proof.
Let us check that α is well defined. If ( H , F ) is an object of sHF ,then α (( H , F )) = N ( H , F ) is a semi-Nelson algebra due to Lemma 3.4.Now we take a morphism f : ( H , F ) −→ ( H ′ , F ′ ). Since f is a Heytinghomomorphism, we have by straightforward calculations that: α ( f )(( a, b ) → ( c, d )) = ( f ( a ⇒ c ) , f ( a ∧ d )) = ( f ( a ) , f ( b )) → (( f ( c ) , f ( d ))) = α ( f )( a, b ) → α ( f )( c, d ), and similarly for the rest of the operations.For all ( a, b ) ∈ N ( H, F ), ( a, b ) = ( Id ( H,F ) ( a ) , Id ( H,F ) ( b ))), therefore α ( Id ( H ,F ) ) = Id α ( H ,F ) .Finally, for any two given morphisms f : ( H ′ , F ′ ) −→ ( H ′′ , F ′′ ), and g : ( H , F ) −→ ( H ′ , F ′ ), we have that for all ( a, b ) ∈ N ( H, F ), α ( f ◦ g )( a, b ) =(( f ◦ g )( a ) , ( f ◦ g )( b )) = ( f ( g ( a )) , f ( g ( b ))) = α ( f )(( g ( a ) , g ( b ))) = (( α ( f )) ◦ ( α ( g )))( a, b ). (cid:4) Definition 3.9
Let A ∈ SN . We say that an element a ∈ A is positive if ∼ a ≤ a . We denote by A + the set of positive elements of A . Observation 3.10 If A ∈ SN , then A + = { x ∨ ∼ x : x ∈ A } . Indeed, if x ∈ A + , then ∼ x ≤ x , so x = x ∨ ∼ x . On the other hand, if y = x ∨ ∼ x , by SN
4) and SN
5) it follows that y = x ∨ ∼ x ≥ x ∧ ∼ x = ∼ ( ∼ x ∨ x ) = ∼ y . We write [[ A + ]] for the set { [[ a ]] : a ∈ A + } . Lemma 3.11 If A ∈ SN , then [[ A + ]] is an i -filter of sH ( A ) . roof. By Theorem 2.12, there exists a subalgebra S of V k ( sH ( A )) suchthat A is isomorphic to S , and the isomorphism is given by h ( a ) = ([[ a ]] , [[ ∼ a ]]) for all a ∈ A . Therefore S = { ([[ a ]] , [[ ∼ a ]]) : a ∈ A } . Then, it is immediatethat π ( S ) = sH ( A ).By Lemma 3.5, S = N ( sH ( A ) , E ), where E = { [[ a ]] ∨ [[ ∼ a ]] : a ∈ A } is an i -filter of sH ( A ). But E = [[ A + ]], due to observation 3.10, which completesthe proof. (cid:4) Proposition 3.12
The application β : sN −→ sHF defined on the ob-jects of sN by β ( A ) = ( sH ( A ) , [[ A + ]]) , and on morphisms f : A −→ A ′ by β ( f )([[ a ]]) = [[ f ( a )]] for all a ∈ A , is a functor from sN to sHF . Proof.
Let us check that β is well defined. For an object A of sN , sH ( A )is a semi-Heyting algebra, and by Lemma 3.11 we have that [[ A + ]] is an i -filterof sH ( A ). Therefore, ( sH ( A ) , [[ A + ]]) is an object in sHF .For any semi-Nelson morphism f : A −→ A ′ , we calculate: β ( f )([[ a → b ]]) = [[ f ( a → b )]] = [[ f ( a ) → f ( b )]] = [[ f ( a )]] ⇒ [[ f ( b )]] = β ( f )([[ a ]]) ⇒ β ( f )([[ b ]]), and similarly for the other operations.Also, if [[ x ]] ∈ [[ A + ]], then by observation 3.10 we can write [[ x ]] = [[ y ∨ ∼ y ]],for some y ∈ A , and so β ( f )([[ x ]]) = β ( f )([[ y ∨ ∼ y ]]) = β ( f )([[ y ]]) ∪ β ( f )([[ ∼ y ]]) = [[ f ( y )]] ∪ [[ f ( ∼ y )]] = [[ f ( y )]] ∪ [[ ∼ f ( y )]] = [[ f ( y ) ∨ ∼ f ( y )]]. Since f ( y ) ∈ f ( A ) ⊆ A ′ , it follows that β ( f )([[ x ]]) ∈ [[ A ′ + ]]. This shows that β ( f )([[ A + ]]) ⊆ [[ A ′ + ]], so β ( f ) is a morphism in sHF .It is straightforward to check that β preserves identities and compositions,so it is indeed a functor. (cid:4) The connection between theorems 3.8 and 3.12 is the following:
Theorem 3.13
The functors α and β establish an equivalence between thecategories sHF and sN . Proof.
Notice that βα ( A , F ) = β ( N ( A , F )) = ( sH ( N ( A , F )) , [[ N ( A, F ) + ]]), with sH ( N ( A , F )) = { [[( x, y )]] : ( x, y ) ∈ N ( A, F ) } = { [[( x, y )]] : x, y ∈ A, x ∧ y =0 , x ∨ y ∈ F } .We define now η A ( x ) = [[( x, x ∗ )]], for each object A of sHF and for all x ∈ A , and we prove that η = { η A } is a natural isomorphism from id sHF to βα . η A is well defined: Since x ∧ x ∗ = 0 and also D s ( A ) ⊆ F it follows that x ∨ x ∗ ∈ D s ( A ) ⊆ F . Hence, ( x, x ∗ ) ∈ N ( A, F ).13 A is surjective: If [[( x, y )]] ∈ βα ( A, F ), then [[( x, y )]] = [[( x, x ∗ )]]. Indeed,since ( x, y ) → ( x, x ∗ ) = ( x ⇒ x, x ∧ x ∗ ) = (1 ,
0) and ( x, x ∗ ) → ( x, y ) = ( x ⇒ x, x ∧ y ) = (1 , x, y )]] = [[( x, x ∗ )]]. Therefore η A ( x ) = [[( x, y )]]. η A is injective: If [[( x, x ∗ )]] = [[( y, y ∗ )]], then ( x, x ∗ ) → ( y, y ∗ ) = ( x ⇒ y, x ∧ y ∗ ) = (1 , x ⇒ y = 1, using properties of semi-Heyting algebras we obtain x ≤ y . Analogously, ( y, y ∗ ) → ( x, x ∗ ) = ( y ⇒ x, y ∧ x ∗ ) = (1 ,
0) implies y ≤ x . η A is a sHF -morphism: • η A ( x ) ∪ η A ( y ) = [[( x, x ∗ )]] ∪ [[( y, y ∗ )]] = [[( x, x ∗ ) ⊔ ( y, y ∗ )]] = [[( x ∨ y, x ∗ ∧ y ∗ )]] = [[( x ∨ y, ( x ∨ y ) ∗ )]] = η A ( x ∨ y ). • η A ( x ) ∩ η A ( y ) = [[( x, x ∗ )]] ∩ [[( y, y ∗ )]] = [[( x, x ∗ ) ⊓ ( y, y ∗ )]] = [[( x ∧ y, x ∗ ∨ y ∗ )]]. Also, η A ( x ∧ y ) = [[( x ∧ y, ( x ∧ y ) ∗ )]]. Since ( x ∧ y, x ∗ ∨ y ∗ ) → ( x ∧ y, ( x ∧ y ) ∗ ) = (( x ∧ y ) ⇒ ( x ∧ y ) , ( x ∧ y ) ∧ ( x ∧ y ) ∗ ) = (1 ,
0) and( x ∧ y, ( x ∧ y ) ∗ ) → ( x ∧ y, x ∗ ∨ y ∗ ) = (( x ∧ y ) ⇒ ( x ∧ y ) , ( x ∧ y ) ∧ ( x ∗ ∨ y ∗ )) = (1 , ( x ∧ y ∧ x ∗ ) ∨ ( x ∧ y ∧ y ∗ )) = (1 , η A ( x ) ∩ η A ( y ) = η A ( x ∧ y ). • η A ( x ) ⇒ η A ( y ) = [[( x, x ∗ )]] ⇒ [[( y, y ∗ )]] = [[( x, x ∗ ) ⇒ ( y, y ∗ )]] = [[( x ⇒ y, x ∧ y ∗ )]]. Also, η A ( x ⇒ y ) = [[( x ⇒ y, ( x ⇒ y ) ∗ )]]. Since ( x ⇒ y, x ∧ y ∗ ) → ( x ⇒ y, ( x ⇒ y ) ∗ ) = (( x ⇒ y ) ⇒ ( x ⇒ y ) , ( x ⇒ y ) ∧ ( x ⇒ y ) ∗ ) = (1 ,
0) and ( x ⇒ y, ( x ⇒ y ) ∗ ) → ( x ⇒ y, x ∧ y ∗ ) = (( x ⇒ y ) ⇒ ( x ⇒ y ) , ( x ⇒ y ) ∧ ( x ∧ y ∗ )) = SH ) (1 , x ∧ ( y ∧ y ∗ )) = (1 , x ⇒ y, x ∧ y ∗ )]] = [[( x ⇒ y, ( x ⇒ y ) ∗ )]]. Therefore, we conclude that η A ( x ) ⇒ η A ( y ) = η A ( x ⇒ y ). • η A ( F ) ⊆ [[ N ( A, F ) + ]]If y ∈ η A ( F ), then y = η A ( x ), for some x ∈ F . Hence, y = [[( x, x ∗ )]],with x ∈ F .We observe that for x ∈ F , we have that ( x, ∈ N ( A, F ). Besides,[[( x, x ∗ )]] = [[( x, x, → ( x, x ∗ ) = ( x ⇒ x, x ∧ x ∗ ) = (1 , x, x ∗ ) → ( x,
0) = ( x ⇒ x, x ∧
0) = (1 , y = [[( x, ∼ ( x,
0) = (0 , x ) ≤ ( x, x, ∈ N ( A, F ) + , and hence y ∈ [[ N ( A, F ) + ]].14inally, we check the naturality of η , that is, the following diagram commutes: A η A / / f (cid:15) (cid:15) βα ( A ) βα ( f ) (cid:15) (cid:15) A ′ η A ′ / / βα ( A ′ )Notice that since η A is an isomorphism, η − A ([[( x, y )]]) = x for every x ∈ A .Then we have ( η − A ′ ◦ βα ( f ) ◦ η A )( x ) = η − A ′ ( βα ( f )([[( x, η − A ′ ([[ α ( f )( x, η − A ′ ([[ f ( x ) , f ( x ) . Thus f = η − A ′ ◦ βα ( f ) ◦ η A , and therefore η A ′ ◦ f = βα ( f ) ◦ η A .Now we consider the functor αβ , which transforms every object A of sN into N ( sH ( A ) , [[ A + ]]). On a morphism f : A −→ A ′ it is defined by αβ ( f )([[ x ]] , [[ y ]]) = ([[ f ( x )]] , [[ f ( y )]]). We define for each object A of sN , δ A : A −→ N ( sH ( A ) , [[ A + ]]) by δ A ( x ) = ([[ x ]] , [[ ∼ x ]]). δ A is well defined: On the one hand, [[ x ]] ∩ [[ ∼ x ]] = [[ x ∧ ∼ x ]] = [[0]],because ( x ∧ ∼ x ) → x ∧ ∼ x ) → ( x ∧ ∼ x ∧
0) = ( x ∧ ∼ x ) → N → N ( x ∧ ∼ x ) = 1 by well known properties of Nelson algebras. On theother hand, using SN
5) and De Morgan properties we have ∼ ( x ∨ ∼ x ) = ∼ x ∧ x ≤ x ∨ ∼ x , so x ∨ ∼ x ∈ A + . Therefore [[ x ]] ∪ [[ ∼ x ]] = [[ x ∨ ∼ x ]] ∈ [[ A + ]]. δ A is surjective: If ([[ x ]] , [[ y ]]) ∈ αβ ( A, F ), then ([[ x ]] , [[ y ]]) = ([[ x ]] , [[ ∼ x ]]).Indeed, this is a consequence of ([[ x ]] , [[ y ]]) → ([[ x ]] , [[ ∼ x ]]) = ([[ x ]] ⇒ [[ x ]] , [[ x ]] ∩ [[ ∼ x ]]) = ([[1]] , [[ x ∧ ∼ x ]]) = ([[1]] , [[0]]) and ([[ x ]] , [[ ∼ x ]]) → ([[ x ]] , [[ y ]]) = ([[ x ]] ⇒ [[ x ]] , [[ x ]] ∩ [[ y ]]) = ([[1]] , [[0]]). Therefore δ A ( x ) = ([[ x ]] , [[ y ]]). δ A is injective: If ([[ x ]] , [[ ∼ x ]]) = ([[ y ]] , [[ ∼ y ]]), then [[ x ]] = [[ y ]] and [[ ∼ x ]] =[[ ∼ y ]], so we have the following: x → y = 1, y → x = 1, ∼ x →∼ y =1, ∼ y →∼ x = 1. By proposition 2.11 (1), we have that x → N y = 1, y → N x = 1, ∼ x → N ∼ y = 1, ∼ y → N ∼ x = 1, which implies (see forexample [Vig99], (1.15)) that x = y . δ A is a sN -morphism: • δ A ( x ) ⊔ δ A ( y ) = ([[ x ]] , [[ ∼ x ]]) ⊔ ([[ y ]] , [[ ∼ y ]]) = ([[ x ]] ∪ [[ y ]] , [[ ∼ x ]] ∩ [[ ∼ y ]]) =([[ x ∨ y ]] , [[ ∼ x ∧ ∼ y ]]) = ([[ x ∨ y ]] , [[ ∼ ( x ∨ y )]]) = δ A ( x ∨ y ). In a similarmanner we can prove that δ A ( x ) ⊓ δ A ( y ) = δ A ( x ∧ y ). • ∼ δ A ( x ) = ∼ ([[ x ]] , [[ ∼ x ]]) = ([[ ∼ x ]] , [[ x ]]) = δ A ( ∼ x ).15 δ A ( x ) → δ A ( y ) = ([[ x ]] , [[ ∼ x ]]) → ([[ y ]] , [[ ∼ y ]]) = ([[ x ]] ⇒ [[ y ]] , [[ x ]] ∩ [[ ∼ y ]]) = ([[ x → y ]] , [[ x ∧ ∼ y ]]).By SN
10) and SN x ∧ ∼ y ]] = [[ ∼ ( x → y )]]. Hence δ A ( x ) → δ A ( y ) = ([[ x → y ]] , [[ x ∧ ∼ y ]]) = ([[ x → y ]] , [[ ∼ ( x → y )]]) = δ A ( x → y ).Finally, δ is a natural transformation: to see that this diagram commutes, A δ A / / f (cid:15) (cid:15) αβ ( A ) αβ ( f ) (cid:15) (cid:15) A ′ δ A ′ / / αβ ( A ′ )we calculate αβ ( f ) ◦ δ A ( x ) = αβ ( f )([[ x ]] , [[ ∼ x ]]) = ([[ f ( x )]] , [[ f ( ∼ x )]]) =([[ f ( x )]] , [[ ∼ f ( x )]]) = δ A ′ ◦ f ( x ). (cid:4) Example 3.14
Consider the semi-Nelson algebra A , with Hasse diagramindicated below, in which the operations → and ∼ are given by the followingtables: ❡ ❡ ❡ ❡❡❡ ❡❡❡ a bd c ef g (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)❅❅❅ ❅❅❅ ❅❅❅❅❅❅ ❅❅❅ ❅❅❅ A → a b c d e f g
10 1 1 1 1 g g ga g g gb g g gc g g gd e g e g e g e d d f f a c gf e g e g e g g d d f f a c g a b c d e f g x a b c d e f g ∼ x g f c e d b a S = { , a, d, e, g, } is the universe of a subalgebra S of A . Both S and A have the same quotient semi-Nelson algebra H indicated below: ❡ ❡❡ [[0]][[ d ]] [[ e ]][[1]] (cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)❅❅❅ ❅❅❅ H ⇒ [[0]] [[ d ]] [[ e ]] [[1]][[0]] [[1]] [[ e ]] [[1]] [[ e ]][[ d ]] [[ e ]] [[1]] [[ e ]] [[1]][[ e ]] [[ d ]] [[0]] [[1]] [[ e ]][[1]] [[0]] [[ d ]] [[ e ]] [[1]] E = { [[ e ]] , [[1]] } is an i-filter of H , and N ( H , E ) is isomorphic to S , while N ( H , H ) = V k ( H ) is isomorphic to A . In this section we will show some results about the variety of dually hemi-morphic semi-Heyting and semi-Nelson algebras, which were presented indefinitions 2.8 and 2.9 respectively. We omit some proofs of the followingresults, since it can be founded in [CSM19].
Lemma 4.1
Let A ∈ DSN . If we consider the equivalence relation ≡ de-fined in section 2.2, then that relation is compatible with respect of the oper-ation ′ . Hence, ≡ is a congruence in DSN . In view of the previous lemma, we can define[[ x ]] † = [[ x ′ ]]and moreover, it can be proved that dsH ( A ) = h A/ ≡ ; ∩ , ∪ , ⇒ , † , ⊥ , ⊤i ∈DSH . Definition 4.2
Let A = h A ; ∩ , ∪ , ⇒ , † , ⊥ , ⊤i ∈ DSH . For ( a, b ) ∈ V k ( A ) we define the following unary operation on V k ( A ) : ( a, b ) ′ = ( a † , b ∩ ( a † ⇒ a )) . The operation is well defined. Indeed, using SH
2) we have a † ∩ b ∩ ( a † ⇒ a ) = b ∩ a † ∩ ( a † ⇒ a ) = b ∩ a † ∩ a = 0.17 heorem 4.3 If A ∈ DSH , then the system V k ( A ) = h V k ( A ); ⊓ , ⊔ , → , ∼ , ′ , ⊤i is a dually hemimorphic semi-Nelson algebra. In a similar manner as on section 2.1, we have the following representa-tions:
Theorem 4.4 [CSM19] If A ∈ DSH , then A is isomorphic to dsH ( V k ( A )) .Also, if A ∈ DSN , then A is isomorphic to a subalgebra of V k ( dsH ( A )) . Now we will generalize the results of section 3. In [CSM18] the authorsintroduced the category SN c of centered semi-Nelson algebras whose objectsare algebras A = h A ; ∧ , ∨ , → , ∼ , c, i of type (2 , , , , ,
0) such that c = ∼ c ,and the morphisms are the algebra homomorphisms. The element c is calleda center of the algebra A , and it is necessarily unique. We can expand thecategory SN c to DSN c of dually hemimorphic centered semi-Nelson algebraswhose objects are algebras A = h A ; ∧ , ∨ , → , ∼ , ′ , c, i of type (2 , , , , , , c = ∼ c , and the morphisms are the algebra homomorphisms.We will prove that the categories DSN and
DSN c actually coincide. Proposition 4.5 If A ∈ DSN , then A is centered, with center ′ . Proof.
We want to prove that ∼ ′ = 1 ′ , or equivalently, the followingconditions holds simultaneously: ∼ ′ → ′ = 1 , ′ →∼ ′ = 1 , ∼ ′ →∼ ( ∼ ′ ) = 1 , ∼ ( ∼ ′ ) →∼ ′ = 1 . (4.1)By Proposition 2.11 (1), (4.1) is equivalent to ∼ ′ → N ′ = 1 , ′ → N ∼ ′ = 1 , ∼ ′ → N ∼ ( ∼ ′ ) = 1 , ∼ ( ∼ ′ ) → N ∼ ′ = 1 , which is equivalent to show that ∼ ′ → N ′ = 1 , (4.2)1 ′ → N ∼ ′ = 1 . (4.3)By (DSN4) , (DSN5) , and proposition 1, we have that ∼ ′ → N ( ∼ ∧ (1 ′ → ∼ ′ → N → N x = x holds for all x , it follows that (2) 0 → N ′ = 1 ′ . By (1), (2) andproposition 2.11 (2), we have that ∼ ′ → N ′ = 1, which proves (4.2).By (DSN2) we have (3) 1 = 1 ′ →∼ ′ → ′ → (1 ′ ∧
0) =1 ′ → N
0. Using again that 0 → N ′ = 1 ′ , and proposition 2.11 (2), gives us ∼ ′ → N (cid:4) Proposition 4.5 shows us that the category
DSN it can be viewed as thecategory
DSN c with a distinguished element c , but despite the change ofthe language, both categories are essentially the same.It was proved in [CSM18] that there is a categorial equivalence betweensemi-Heyting algebras and centered semi-Nelson algebras. Moreover, thefunction h defined in the proof of Theorem 4.4 is an isomorphism. This givesus the following: Theorem 4.6 If A ∈ DSN , then A is isomorphic to V k ( dsH ( A )) . Observation 4.7
Another way of proving Theorem 4.6 is the following: ifwe consider the structure N ( H , E ) as in Lemma 3.5, and take H ∈ DSH ,we have no guarantee that this set is going to be a closed under the operation † . If instead we consider a subalgebra S such that π ( S ) = H , then theproof indicated in the lemma still holds for the variety DSH , since it doesnot use the operations ′ or † . Hence, the sets S and N ( H, E ) are equal, so N ( H, E ) inherits the operations from S . Indeed, due to Theorem 4.4, weknow that the function h : A → V k ( dsH ( A )) defined by h ( a ) = ([[ a ]] , [[ ∼ a ]]) is a monomorphism. Hence, A ∼ = h ( A ) . Let us take then the subalgebra S = h ( A ) of V k ( dsH ( A )) . It is immediate that π ( S ) = dsH ( A ) . Therefore,by Lemma 3.5, h ( A ) = N ( dsH ( A ) , E ) for E = { [[ x ]] ∈ dsH ( A ) : [[ x ]] =[[ a ]] ∨ [[ b ]] , ([[ a ]] , [[ b ]]) ∈ S } .Since S is a subalgebra of V k ( dsH ( A )) , we have that ([[1]] , [[0]]) ∈ S , sothe center ([[0]] , [[0]]) = ([[1]] , [[0]]) ′ is an element of S . Then [[0]] = [[0]] ∨ [[0]] ∈ E .Since E is a filter, this shows that E = dsH ( A ) .We have proved that S = N ( dsH ( A ) , dsH ( A )) = V k ( dsH ( A )) , andtherefore h is an isomorphism. Example 4.8
Consider the semi-Nelson algebras A and its subalgebra S from Example 3.14. We know by the results above that S cannot be expandedby adding a unary operation ′ in such a way that the system B = h B ; ∧ , ∨ , → ∼ , ′ , i becomes a dually hemimorphic semi-Nelson algebra, since S has nocenter.On the other hand, any operation ′ that makes sH ( A ) into a dually hemi-morphic semi-Heyting algebra yields an operation † that will turn A into adually hemimorphic semi-Nelson algebra. References [ACDV13] Manuel Abad, Juan Manuel Cornejo, and Jose PatricioDiaz Varela. Semi-heyting algebras term-equivalent to g¨odel al-gebras.
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