Adjoint operations in twist-products of lattices
aa r X i v : . [ m a t h . R A ] J a n Adjoint operations in twist-products of lattices
Ivan Chajda and Helmut L¨anger
Abstract
Given an integral commutative residuated lattice L = ( L, ∨ , ∧ ), its full twist-product ( L , ⊔ , ⊓ ) can be endowed with two binary operations ⊙ and ⇒ introducedformerly by M. Busaniche and R. Cignoli as well as by C. Tsinakis and A. M. Willesuch that it becomes a commutative residuated lattice. For every a ∈ L we definea certain subset P a ( L ) of L . We characterize when P a ( L ) is a sublattice of the fulltwist-product ( L , ⊔ , ⊓ ). In this case P a ( L ) together with some natural antitone in-volution ′ becomes a pseudo-Kleene lattice. If L is distributive then ( P a ( L ) , ⊔ , ⊓ , ′ )becomes a Kleene lattice. We present sufficient conditions for P a ( L ) being a subal-gebra of ( L , ⊔ , ⊓ , ⊙ , ⇒ ) and thus for ⊙ and ⇒ being a pair of adjoint operationson P a ( L ). Finally, we introduce another pair ⊙ and ⇒ of adjoint operations onthe full twist-product of a bounded commutative residuated lattice such that theresulting algebra is a bounded commutative residuated lattice satisfying the doublenegation law and we investigate when P a ( L ) is closed under these new operations ⊙ and ⇒ . AMS Subject Classification:
Keywords:
Full twist-product, residuated lattice, Kleene lattice, pseudo-Kleene lattice,double negation law
Kleene lattices were introduced by J. A. Kalman ([6]) (under a different name) as aspecial kind of De Morgan lattices which serve as an algebraic axiomatization of a certainpropositional logic satisfying the double negation law but not necessary the excludedmiddle law. If the underlying lattice is not distributive such lattices are called pseudo-Kleene (see e.g. [3]). It is a question if certain binary operations can be introduced ina Kleene or pseudo-Kleene lattice such that they form an adjoint pair. To solve thisproblem, we apply an approach using the full twist-product construction and anotherconstruction extending a distributive lattice into a Kleene one.Having a residuated lattice ( L, ∨ , ∧ , · , → , ⊙ and ⇒ on the fulltwist-product ( L , ⊔ , ⊓ ) to be converted into a residuated lattice ( L , ⊔ , ⊓ , ⊙ , ⇒ , (1 , Support of the research by the Austrian Science Fund (FWF), project I 4579-N, and the CzechScience Foundation (GA ˇCR), project 20-09869L, entitled “The many facets of orthomodularity”, as wellas by ¨OAD, project CZ 02/2019, entitled “Function algebras and ordered structures related to logic anddata fusion”, and, concerning the first author, by IGA, project PˇrF 2020 014, is gratefully acknowledged.
1t is known that if L = ( L, ∨ , ∧ , ′ ) is a distributive lattice with an antitone involution, a ∈ L and P a ( L ) := { ( x, y ) ∈ L | x ∧ y ≤ a ≤ x ∨ y } then ( P a ( L ) , ∨ , ∧ , ′ ) is a Kleenelattice. If L is not distributive then the situation is different.Our aim is to combine both of these approaches and hence ask for several questions asfollows: • When is ( P a ( L ) , ⊔ , ⊓ ) a sublattice of the full twist-product ( L , ⊔ , ⊓ ), also in thecase of a non-distributive lattice L ? • When is P a ( L ) closed under operations ⊙ and ⇒ mentioned above? • When can P a ( L ) be equipped with these operations forming an adjoint pair? • Can we define the operations ⊙ and ⇒ in a way different from that of [2] or [7] toobtain an integral residuated lattice on the full twist-product ( L , ⊔ , ⊓ )?We answer these question in our paper by giving sufficient and, in some cases, also nec-essary conditions under which we get a positive solution. Moreover, we present examplesshowing how our constructions work. We recall several concepts which will be used throughout the paper. Moreover, we recallsome results already published on which our present study is based.Let P = ( P, ≤ ) be a poset. An antitone involution on P is a unary operation ′ on P satisfying(i) x ≤ y implies y ′ ≤ x ′ ,(ii) x ′′ ≈ x for all x, y ∈ P . A distributive lattice having an antitone involution is called a De Morganlattice or a
De Morgan algebra . Definition 2.1. A commutative residuated lattice is an algebra ( L, ∨ , ∧ , · , → , of type (2 , , , , such that (i) ( L, ∨ , ∧ ) is a lattice, (ii) ( L, · , is a commutative monoid, (iii) for all x, y, z ∈ L , x · y ≤ z is equivalent to x ≤ y → z (adjointness property) . ( L, ∨ , ∧ , · , → , is called integral if is the top element of the lattice ( L, ∨ , ∧ ) . A com-mutative residuated lattice with 0 is an algebra ( L, ∨ , ∧ , · , → , , of type (2 , , , , , such that ( L, ∨ , ∧ , · , → , is a commutative residuated lattice and is the bottom elementof ( L, ∨ , ∧ ) . Let ( L, ∨ , ∧ , · , → , , be a commutative residuated lattice with . Define x ′ := x → for all x ∈ L . ( L, ∨ , ∧ , · , → , , is called a bounded commutative residuated lattice if is the top element of ( L, ∨ , ∧ ) , • said to satisfy the double negation law if it satisfies the identity x ′′ ≈ x , i.e. ( x → → ≈ x . We say that the operations · and → form an adjoint pair if they satisfy the adjointness(iii) of Definition 2.1.The following properties of integral commutative residuated lattices are well-known (cf.e.g. [1]). Proposition 2.2.
Let ( L, ∨ , ∧ , · , → , be an integral commutative residuated lattice.Then the following hold for all x, y, z ∈ L : (i) x ≤ y implies x · z ≤ y · z , (ii) x · y ≤ x, y , (iii) 1 → x ≈ x , (iv) x ≤ y → x , (v) x → y = 1 if and only if x ≤ y , (vi) x ≤ y implies y → z ≤ x → z , (vii) x ≤ y implies z → x ≤ z → y , (viii) x → ( y ∧ z ) ≈ ( x → y ) ∧ ( x → z ) , (ix) ( x · y ) → z ≈ x → ( y → z ) . Let L = ( L, ∨ , ∧ ) be a lattice. By the full twist-product of L is meant the lattice ( L , ⊔ , ⊓ )where ⊔ and ⊓ are defined as follows:( x, y ) ⊔ ( z, v ) := ( x ∨ z, y ∧ v ) , ( x, y ) ⊓ ( z, v ) := ( x ∧ z, y ∨ v )for all ( x, y ) , ( z, v ) ∈ L . Hence ( x, y ) ≤ ( z, v ) if and only if both x ≤ z and v ≤ y . Assumenow that ( L, ∨ , ∧ , · , → ,
1) is an integral commutative residuated lattice. In Theorem 3.1in [2] which is a particular case of Corollary 3.6 in [7], Busaniche and Cignoli introducedtwo additional binary operations ⊙ and ⇒ on its full twist-product ( L , ⊔ , ⊓ ) as follows:( x, y ) ⊙ ( z, v ) := ( x · z, ( x → v ) ∧ ( z → y )) , (1)( x, y ) ⇒ ( z, v ) := (( x → z ) ∧ ( v → y ) , x · v ) (2)for all ( x, y ) , ( z, v ) ∈ L . They showed that ( L , ⊔ , ⊓ , ⊙ , ⇒ , (1 , ⊙ and ⇒ form an adjoint pair. For the convenience of the readerwe provide a proof since it is not explicitly contained in [2] and [7]. Theorem 2.3.
Let L = ( L, ∨ , ∧ , · , → , be an integral commutative residuated latticeand ⊙ and ⇒ defined by (1) and (2) , respectively. Then ( L , ⊔ , ⊓ , ⊙ , ⇒ , (1 , is acommutative residuated lattice. roof. Let a, b, c, d, e, f ∈ L .(i) It is easy to see that ( L , ⊔ , ⊓ ) is a lattice.(ii) We prove that ( L , ⊙ , (1 , x, y ) ⊙ ( z, v ) ≈ ( x · z, ( x → v ) ∧ ( z → y )) ≈ ( z · x, ( z → y ) ∧ ( x → v )) ≈≈ ( z, v ) ⊙ ( x, y ) , (( x, y ) ⊙ ( z, v )) ⊙ ( t, w ) ≈ ( x · z, ( x → v ) ∧ ( z → y )) ⊙ ( t, w ) ≈≈ (( x · z ) · t, (( x · z ) → w ) ∧ ( t → (( x → v ) ∧ ( z → y )))) ≈≈ ( x · ( z · t ) , (( x · z ) → w ) ∧ ( t → ( x → v )) ∧ ( t → ( z → y ))) ≈≈ ( x · ( z · t ) , ( x → ( z → w )) ∧ ( x → ( t → v )) ∧ (( z · t ) → y )) ≈≈ ( x · ( z · t ) , ( a → (( z → w ) ∧ ( t → v ))) ∧ (( z · t ) → y )) ≈≈ ( x, y ) ⊙ ( z · t, ( z → w ) ∧ ( t → v ) ≈≈ ( x, y ) ⊙ (( z, v ) ⊙ ( t, w )) , ( x, y ) ⊙ (1 , ≈ ( x · , ( x → ∧ (1 → y )) ≈ ( x, ∧ y ) ≈ ( x, y ) . (iii) Now we prove the adjointness property. The following are equivalent:( a, b ) ⊙ ( c, d ) ≤ ( e, f ) , ( a · c, ( a → d ) ∧ ( c → b )) ≤ ( e, f ) ,a · c ≤ e and f ≤ ( a → d ) ∧ ( c → b ) ,a · c ≤ e, f ≤ a → d and f ≤ c → b,a ≤ c → e, a ≤ f → d and c · f ≤ b,a ≤ ( c → e ) ∧ ( f → d ) and c · f ≤ b, ( a, b ) ≤ (( c → e ) ∧ ( f → d ) , c · f ) , ( a, b ) ≤ ( c, d ) ⇒ ( e, f ) . It is worth noticing that the operations ⊙ and ⇒ defined above are not independent.Namely one can be expressed by the other by using the antitone involution ′ defined by( x, y ) ′ := ( y, x ). Namely,( x, y ) ⊙ ( z, v ) ≈ ( x · z, ( x → v ) ∧ ( z → y ) ≈ (( x → v ) ∧ ( z → y ) , x · z ) ′ ≈≈ (( x, y ) ⇒ ( v, z )) ′ ≈ (( x, y ) ⇒ ( z, v ) ′ ) ′ , ( x, y ) ⇒ ( z, v ) ≈ (( x, y ) ⇒ ( z, v ) ′′ ) ′′ ≈ (( x, y ) ⊙ ( z, v ) ′ ) ′ . Moreover, note that the residuated lattice ( L , ⊔ , ⊓ , ⊙ , ⇒ , (1 , ,
0) of the full twist-product is different from the neutralelement (1 ,
1) of the monoid ( L , ⊙ , (1 , pseudo-Kleene lattice is an algebra ( L, ∨ , ∧ , ′ ) of type (2 , ,
1) such that the followinghold for all x, y ∈ L : 4i) L = ( L, ∨ , ∧ ) is a lattice,(ii) ′ is an antitone involution on ( L, ≤ ),(iii) x ∧ x ′ ≤ y ∨ y ′ .(Here and in the rest of the paper ≤ denotes the induced order of the lattice L .) If,moreover, L is distributive then ( L, ∨ , ∧ , ′ ) is called a Kleene lattice . Let L = ( L, ∨ , ∧ ) be a lattice and ( L , ⊔ , ⊓ ) its full twist-product. It is easy to see that( L , ⊔ , ⊓ ) is distributive if and only if so is L . The following construction was introducedfor distributive lattices in [5] and generalized for posets by the authors in [4]: Let a ∈ L and consider the following subset of L : P a ( L ) := { ( x, y ) ∈ L | x ∧ y ≤ a ≤ x ∨ y } . Since our paper [4] is devoted to posets and not to lattices, we are going to show that if( P a ( L ) , ⊔ , ⊓ ) is a sublattice of ( L , ⊔ , ⊓ ) then ( P a ( L ) , ⊔ , ⊓ , ′ ) where the unary operation ′ on P a ( L ) is defined by ( x, y ) ′ := ( y, x ) for all ( x, y ) ∈ P a ( L ) is a pseudo-Kleene lattice. Theorem 3.1.
Let L = ( L, ∨ , ∧ ) be a lattice and a ∈ L , assume that ( P a ( L ) , ⊔ , ⊓ ) is asublattice of ( L , ⊔ , ⊓ ) and put ( x, y ) ′ := ( y, x ) for all ( x, y ) ∈ L . Then (i) ( P a ( L ) , ⊔ , ⊓ , ′ ) is a pseudo-Kleene lattice, (ii) the mapping x ( x, a ) is an embedding of L into ( P a ( L ) , ⊔ , ⊓ ) , (iii) ( P a ( L ) , ⊔ , ⊓ ) is distributive if and only if so is L .Proof. Let ( b, c ) , ( d, e ) ∈ P a ( L ) and f, g ∈ L .(i) The following are equivalent: ( b, c ) ≤ ( d, e ) ,b ≤ d and e ≤ c,e ≤ c and b ≤ d, ( e, d ) ≤ ( c, b ) , ( d, e ) ′ ≤ ( b, c ) ′ . Further, we have ( b, c ) ′′ = ( c, b ) ′ = ( b, c ). Thus ′ is an antitone involution on( P a ( L ) , ⊔ , ⊓ ). Moreover,( b, c ) ⊓ ( b, c ) ′ = ( b, c ) ⊓ ( c, b ) = ( b ∧ c, c ∨ b ) ≤ ( a, a ) ≤ ( d ∨ e, e ∧ d ) == ( d, e ) ⊔ ( e, d ) = ( d, e ) ⊔ ( d, e ) ′ proving that ( P a ( L ) , ⊔ , ⊓ , ′ ) is a pseudo-Kleene lattice.5ii) Since we have ( f, a ) ≤ ( g, a ) if and only if f ≤ g , it is evident.(iii) This can be easily checked.In general, ( P a ( L ) , ⊔ , ⊓ ) need not be a sublattice of ( L , ⊔ , ⊓ ). Example 3.2.
Consider the lattice N = ( N , ∨ , ∧ ) depicted in Figure 1: ✉✉ ✉✉ ✉ ❆❆❆❆ ✂✂✂✂✂✂✁✁✁✁ ❇❇❇❇❇❇ ab c Then ( a, , ( c, b ) ∈ P a ( N ) , but ( a, ⊔ ( c, b ) = ( a ∨ c, ∧ b ) = (1 , b ) / ∈ P a ( N ) since ∧ b = b a . This shows that P a ( N ) is not a sublattice of the full twist-product ( N , ⊔ , ⊓ ) of N . We can give a necessary and sufficient condition for ( P a ( L ) , ⊔ , ⊓ ) being a sublattice of( L , ⊔ , ⊓ ). Theorem 3.3.
Let L = ( L, ∨ , ∧ ) be a lattice and a ∈ L . Then ( P a ( L ) , ⊔ , ⊓ ) is a sublatticeof ( L , ⊔ , ⊓ ) if and only if the following condition holds for all x, y, z, v ∈ L : ( x ∧ y ) ∨ ( z ∧ v ) ≤ a ≤ ( x ∨ y ) ∧ ( z ∨ v ) implies (( x ∨ z ) ∧ y ∧ v ) ∨ ( x ∧ z ∧ ( y ∨ v )) ≤ a ≤ ( x ∨ z ∨ ( y ∧ v )) ∧ (( x ∧ z ) ∨ y ∨ v ) . Proof.
Let b, c, d, e ∈ L .The following are equivalent:( b, c ) , ( d, e ) ∈ P a ( L ) ,b ∧ c ≤ a ≤ b ∨ c and d ∧ e ≤ a ≤ d ∨ e, ( b ∧ c ) ∨ ( d ∧ e ) ≤ a ≤ ( b ∨ c ) ∧ ( d ∨ e ) . Moreover, the following are equivalent:( b, c ) ⊔ ( d, e ) ∈ P a ( L ) , ( b ∨ d, c ∧ e ) ∈ P a ( L ) , ( b ∨ d ) ∧ c ∧ e ≤ a ≤ b ∨ d ∨ ( c ∧ e ) . b, c ) ⊓ ( d, e ) ∈ P a ( L ) , ( b ∧ d, c ∨ e ) ∈ P a ( L ) ,b ∧ d ∧ ( c ∨ e ) ≤ a ≤ ( b ∧ d ) ∨ c ∨ e. Corollary 3.4.
Let L = ( L, ∨ , ∧ ) be a distributive lattice and a ∈ L . Then ( P a ( L ) , ⊔ , ⊓ ) is a sublattice of ( L , ⊔ , ⊓ ) and ( P a ( L ) , ⊔ , ⊓ , ′ ) where the antitone involution is given by ( x, y ) ′ := ( y, x ) for all ( x, y ) ∈ P a ( L ) is a Kleene lattice.Proof. If b, c, d, e ∈ L and( b ∧ c ) ∨ ( d ∧ e ) ≤ a ≤ ( b ∨ c ) ∧ ( d ∨ e )then(( b ∨ d ) ∧ c ∧ e ) ∨ ( b ∧ d ∧ ( c ∨ e )) = ( b ∧ c ∧ e ) ∨ ( d ∧ c ∧ e ) ∨ ( b ∧ d ∧ c ) ∨ ( b ∧ d ∧ e ) ≤≤ ( b ∧ c ) ∨ ( d ∧ e ) ∨ ( b ∧ c ) ∨ ( d ∧ e ) ≤ a ≤≤ ( b ∨ c ) ∧ ( d ∨ e ) ∧ ( b ∨ c ) ∧ ( d ∨ e ) ≤≤ ( b ∨ d ∨ c ) ∧ ( b ∨ d ∨ e ) ∧ ( b ∨ c ∨ e ) ∧ ( d ∨ c ∨ e ) ≤≤ ( b ∨ d ∨ ( c ∧ e )) ∧ (( b ∧ d ) ∨ c ∨ e ) . The rest of proof follows by Theorem 3.1.The following example shows a distributive lattice L having an element a such that( P a ( L ) , ⊔ , ⊓ ) is a sublattice of the full twist-product ( L , ⊔ , ⊓ ). Example 3.5.
Consider the lattice L = ( L, ∨ , ∧ ) visualized in Figure 2: ✉✉✉✉ ab If one defines binary operations · and → on L by x · y := x ∧ y and x → y := (cid:26) if x ≤ y,y otherwise , then ( L, ∨ , ∧ , · , → , is an distributive integral commutative residuated lattice. With re-spect to the binary operations ⊙ and ⇒ defined by (1) and (2) , respectively, ( P a ( L ) , ⊔ , ⊓ , ⊙ , ⇒ , (0 , , (1 , is a bounded commutative residuated lattice. According to Corol-lary 3.4, ( P a ( L ) , ⊔ , ⊓ ) is a sublattice if ( L , ⊔ , ⊓ ) . The Hasse diagram of ( P a ( L ) , ⊔ , ⊓ ) isdepicted in Figure 3. ✉ ✉✉ ✉✉✉ ✉✉ ✉✉ ❅❅❅❅ (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)(cid:0)(cid:0) ❅❅❅❅(cid:0)(cid:0)(cid:0)(cid:0) ❅❅❅❅ (0 , , b ) ( a, , a ) ( a, b )( a, a )( b, a ) ( a, , a ) ( b, ,
0) Fig. 3In the following, a special role will play the lattices P a ( L ) all elements of which arecomparable with ( a, a ). We can characterize them as follows. Theorem 3.6.
Let L = ( L, ∨ , ∧ ) be a lattice and a ∈ L . Then the following are equiva-lent: (i) P a ( L ) ⊆ { ( x, y ) ∈ L | ( x, y ) is comparable with ( a, a ) } (ii) P a ( L ) = { ( x, y ) ∈ L | ( x, y ) is comparable with ( a, a ) } (iii) Every element of L is comparable with a and a is ∨ -irreducible and ∧ -irreducible.If this is the case then ( P a ( L ) , ⊔ , ⊓ ) is a sublattice of ( L , ⊔ , ⊓ ) .Proof. Let b, c ∈ L .(i) and (ii) are equivalent since { ( x, y ) ∈ L | ( x, y ) is comparable with ( a, a ) } ⊆ P a ( L ).(i) ⇒ (iii):Since ( a, a ) ≤ ( a, b ) or ( a, b ) ≤ ( a, a ) we have b ≤ a or a ≤ b . If a = b ∨ c then b, c < a would imply ( b, c ) ∈ P a ( L ) and ( b, c ) k ( a, a ), a contradiction. Hence a is ∨ -irreducible.If a = b ∧ c then a < b, c would imply ( b, c ) ∈ P a ( L ) and ( b, c ) k ( a, a ), a contradiction.Hence a is ∧ -irreducible.(iii) ⇒ (i):Let ( b, c ) ∈ P a ( L ). Then b ∧ c ≤ a ≤ b ∨ c .If b = a then ( b, c ) = ( a, c ) is comparable with ( a, a ).If c = a then ( b, c ) = ( b, a ) is comparable with ( a, a ). b, c < a is impossible because of a ≤ b ∨ c . a < b, c is impossible because of b ∧ c ≤ a .If b < a < c then ( b, c ) ≤ ( a, a ). 8f c < a < b then ( a, a ) ≤ ( b, c ).Now assume that (ii) holds and let ( b, c ) , ( d, e ) ∈ P a ( L ).If ( b, c ) , ( d, e ) ≤ ( a, a ) then( b, c ) ⊔ ( d, e ) = ( b ∨ d, c ∧ e ) ≤ ( a, a ) , ( b, c ) ⊓ ( d, e ) = ( b ∧ d, c ∨ e ) ≤ ( a, a ) . If ( b, c ) ≤ ( a, a ) ≤ ( d, e ) then( b, c ) ⊔ ( d, e ) = ( b ∨ d, c ∧ e ) ≥ ( a, a ) , ( b, c ) ⊓ ( d, e ) = ( b ∧ d, c ∨ e ) ≤ ( a, a ) , If ( d, e ) ≤ ( a, a ) ≤ ( b, c ) then( b, c ) ⊔ ( d, e ) = ( b ∨ d, c ∧ e ) ≥ ( a, a ) , ( b, c ) ⊓ ( d, e ) = ( b ∧ d, c ∨ e ) ≤ ( a, a ) . If ( a, a ) ≤ ( b, c ) , ( d, e ) then( b, c ) ⊔ ( d, e ) = ( b ∨ d, c ∧ e ) ≥ ( a, a ) , ( b, c ) ⊓ ( d, e ) = ( b ∧ d, c ∨ e ) ≥ ( a, a ) . Hence ( P a ( L ) , ⊔ , ⊓ ) is a sublattice of ( L , ⊔ , ⊓ ). Example 3.7.
We can see that the lattice L and its element a from Example 3.5 satisfythe conditions of Theorem 3.6 (iii) , hence all elements of P a ( L ) are comparable with theelement ( a, a ) , see Figure 3. P a ( L ) Since the element (1 ,
1) of L does not belong to P a ( L ) unless a = 1, we cannot expectthat ( P a ( L ) , ⊔ , ⊓ ) will be a residuated lattice with respect to operations ⊙ and ⇒ definedby (1) and (2), respectively. On the other hand, it would be important to know when P a ( L ) is closed with respect to ⊙ and ⇒ because then they form an adjoint pair. Hence,if the pseudo-Kleene lattice P a ( L ) represents a certain logic where ⊙ is conjunction and ⇒ is implication then from the trivial inequality x ⇒ y ≤ x ⇒ y we infer by adjointness ( x ⇒ y ) ⊙ x ≤ y, in other words, the propositional value of y is at least as high as the propositional valuesof the conjunction of x ⇒ y and x . This means that this logic satisfies Modus Ponens inthe fuzzy modification and hence this pseudo-Kleene logic enables deduction.Now we are ready to state and prove one of our main results.9 heorem 4.1.
Let ( L, ∨ , ∧ , · , → , be an integral commutative residuated lattice and a an idempotent ( with respect to · ) ∨ -irreducible and ∧ -irreducible element of L which iscomparable with every element of L and put L := ( L, ∨ , ∧ ) . Then ( P a ( L ) , ⊔ , ⊓ , ⊙ , ⇒ ) isa subalgebra of ( L , ⊔ , ⊓ , ⊙ , ⇒ ) and hence ⊙ and ⇒ form an adjoint pair if and only ifthe following two conditions hold for all x, y ∈ L : a · x < a implies a · x = 0 , (3) a < x · y implies ( x → a ) ∧ ( y → a ) = a. (4) Proof.
Let ( b, c ) , ( d, e ) ∈ P a ( L ). According to Theorem 3.6, P a ( L ) = { ( x, y ) ∈ L | ( x, y ) is comparable with ( a, a ) } and we have that ( P a ( L ) , ⊔ , ⊓ ) is a sublattice of ( L , ⊔ , ⊓ ). Since P a ( L ) is closed withrespect to ′ , it is closed with respect to ⇒ if it is closed with respect to ⊙ . Hence, weneed only to check when P a ( L ) is closed with respect to ⊙ .(i) Assume ( b, c ) , ( d, e ) ≤ ( a, a ).Because of (ii) and (iv) of Proposition 2.2 we have( b, c ) ⊙ ( d, e ) = ( b · d, ( b → e ) ∧ ( d → c )) ≤ ( a, a ) . (ii) Assume ( b, c ) ≤ ( a, a ) ≤ ( d, e ).Because of (ii) of Proposition 2.2 we have b · d ≤ a .If b · d = a then ( b, c ) ⊙ ( d, e ) = ( b · d, ( b → e ) ∧ ( d → c )) is comparable with ( a, a ).If b · d < a then ( b, c ) ⊙ ( d, e ) = ( b · d, ( b → e ) ∧ ( d → c )) is comparable with ( a, a )if and only if a ≤ b → e .(iii) Assume ( d, e ) ≤ ( a, a ) ≤ ( b, c ).Because of the commutativity of ⊙ this case reduces to the previous one.(iv) Assume ( a, a ) ≤ ( b, c ) , ( d, e ).Because of (i) of Proposition 2.2 we have a ≤ b · d .If a = b · d then ( b, c ) ⊙ ( d, e ) = ( b · d, ( b → e ) ∧ ( d → c )) is comparable with ( a, a ).If a < b · d then ( b, c ) ⊙ ( d, e ) = ( b · d, ( b → e ) ∧ ( d → c )) is comparable with ( a, a )if and only if ( b → e ) ∧ ( d → c ) ≤ a .Hence ( P a ( L ) , ⊔ , ⊓ , ⊙ , ⇒ ) is a subalgebra of ( L , ⊔ , ⊓ , ⊙ , ⇒ ) if and only if the followingstatements hold:(a) b, e ≤ a ≤ c, d and b · d < a imply a ≤ b → e .(b) c, e ≤ a ≤ b, d and a < b · d imply ( b → e ) ∧ ( d → c ) ≤ a .Because of (i) and (vii) of Proposition 2.2, (a) is equivalent to the following statements: b · a < a implies a ≤ b → ,a · b < a implies a · b ≤ ,a · b < a implies a · b = 0 , (3) . a < b · d implies ( b → a ) ∧ ( d → a ) ≤ a,a < b · d implies ( b → a ) ∧ ( d → a ) = a, (4) . Corollary 4.2.
Let L = ( L, ∨ , ∧ , · , → , be an integral commutative distributive residu-ated lattice and a ∈ L with a · a = a and assume that every element of P a ( L ) is comparablewith ( a, a ) . Then ( P a ( L ) , ⊔ , ⊓ , ′ ) where ( x, y ) ′ := ( y, x ) for all ( x, y ) ∈ P a ( L ) is a Kleenelattice and ⊙ and ⇒ form an adjoint pair if and only if (3) and (4) hold. Example 4.3.
Consider the lattice L = ( L, ∨ , ∧ ) with element a from Example 3.5. Onecan easily check that L satisfies the conditions of Theorem 4.1 and hence ( P a ( L ) , ⊔ , ⊓ , ⊙ , ⇒ ) is a subalgebra of ( L , ⊔ , ⊓ , ⊙ , ⇒ ) . Lemma 4.4.
Let L = ( L, ∨ , ∧ , · , → , be a distributive commutative residuated lat-tice and a ∈ L . Then ( P a ( L ) , ⊔ , ⊓ ) is a distributive sublattice of the full twist-product ( L , ⊔ , ⊓ ) closed with respect to ⊙ ( and hence also with respect to ⇒ ) if and only if forall ( b, c ) , ( d, e ) ∈ P a ( L )( b · d ) ∧ ( b → e ) ∧ ( d → c ) ≤ a ≤ ( b · d ) ∨ ( b → e ) and a ≤ ( b · d ) ∨ ( d → c ) . Proof.
According to Theorem 3.1 and Corollary 3.4, ( P a ( L ) , ⊔ , ⊓ ) is a distributive sub-lattice of ( L , ⊔ , ⊓ ). Let ( b, c ) , ( d, e ) ∈ P a ( L ) and put f := b · d , g := b → e , h := d → c and i := g ∧ h . Then the following are equivalent:( b, c ) ⊙ ( d, e ) ∈ P a ( L ) , ( f, i ) ∈ P a ( L ) ,f ∧ i ≤ a ≤ f ∨ i,f ∧ i ≤ a ≤ f ∨ ( g ∧ h ) ,f ∧ i ≤ a ≤ ( f ∨ g ) ∧ ( f ∨ h ) ,f ∧ i ≤ a ≤ f ∨ g and a ≤ f ∨ h. Corollary 4.5.
Let L = ( L, ∨ , ∧ , · , → , , be a distributive bounded commutative resid-uated lattice and a an atom of L . Then ( P a ( L ) , ⊔ , ⊓ ) is a distributive sublattice of thefull twist-product ( L , ⊔ , ⊓ ) closed with respect to ⊙ ( and hence also with respect to ⇒ ) if and only if for all ( b, c ) , ( d, e ) ∈ P a ( L ) either (i) or (ii) hold: (i) ( b · d ) ∧ ( b → e ) ∧ ( d → c ) = a , (ii) ( b · d ) ∧ ( b → e ) ∧ ( d → c ) = 0 and ( a ≤ b · d or a ≤ ( b → e ) ∧ ( d → c )) .Proof. Let ( b, c ) , ( d, e ) ∈ P a ( L ) and put f := b · d , g := b → e , h := d → c and i := g ∧ h .According to Lemma 4.4, ( P a ( L ) , ⊔ , ⊓ ) is a distributive sublattice of the full twist-product( L , ⊔ , ⊓ ) and ( b, c ) ⊙ ( d, e ) ∈ P a ( L ) is equivalent to ( f ∧ i ≤ a ≤ f ∨ g and a ≤ f ∨ h ).11ow f ∧ i = a implies a ≤ f ∨ i . Using the fact that a is an atom of L we see that thefollowing are equivalent: a ≤ f ∨ g,a ∧ ( f ∨ g ) = a, ( a ∧ f ) ∨ ( a ∧ g ) = a,a ∧ f = a or a ∧ g = a,a ≤ f or a ≤ g. Analogously, a ≤ f ∨ h is equivalent to ( a ≤ f or a ≤ h ). Finally, the following areequivalent:( b, c ) ⊙ ( d, e ) ∈ P a ( L ) ,f ∧ i = a or ( f ∧ i = 0 and ( a ≤ f or a ≤ g ) and ( a ≤ f or a ≤ h )) ,f ∧ i = a or ( f ∧ i = 0 and ( a ≤ f or ( a ≤ g ) and a ≤ h ))) ,f ∧ i = a or ( f ∧ i = 0 and ( a ≤ f or a ≤ g ∧ h )) . Analogously as in Corollary 4.5, we can consider the operation ⇒ instead of ⊙ and provea similar result. Lemma 4.6.
Let L = ( L, ∨ , ∧ , · , → , , be a distributive bounded commutative residu-ated lattice and a an atom of L . Then ( P a ( L ) , ⊔ , ⊓ ) is a distributive sublattice of the fulltwist-product ( L , ⊔ , ⊓ ) closed with respect to ⇒ ( and hence also with respect to ⊙ ) if andonly if for all ( b, c ) , ( d, e ) ∈ P a ( L ) either (i) or (ii) hold: (i) ( b → d ) ∧ ( e → c ) ∧ ( b · e ) = a , (ii) ( b → d ) ∧ ( e → c ) ∧ ( b · e ) = 0 and ( a ≤ ( b → d ) ∧ ( e → c ) or a ≤ b · e ) .Proof. Let ( b, c ) , ( d, e ) ∈ P a ( L ) and put f := b → d , g := e → c , h := b · e and i := f ∧ g .According to Theorem 3.1 and Corollary 3.4, ( P a ( L ) , ⊔ , ⊓ ) is a distributive sublattice ofthe full twist-product ( L , ⊔ , ⊓ ). Now the following are equivalent:( b, c ) ⊙ ( d, e ) ∈ P a ( L ) , ( i, h ) ∈ P a ( L ) ,i ∧ h ≤ a ≤ i ∨ h,i ∧ h = a or ( i ∧ h = 0 and a ≤ i ∨ h ) ,i ∧ h = a or ( i ∧ h = 0 and ( a ≤ i or a ≤ h )) . (that a ≤ i ∨ h is equivalent to ( a ≤ i or a ≤ h ) follows like in the proof of Corollary 4.5). Example 4.7.
Consider the lattice L = ( L, ∨ , ∧ ) shown in Figure 4: ✉ ✉✉ ✉✉ ❅❅❅❅ (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)❅❅❅❅ ❅❅❅❅ b ac d According to Corollary 3.4, ( P a ( L ) , ⊔ , ⊓ ) is a sublattice of the full twist-product ( L , ⊔ , ⊓ ) .The Hasse diagram of ( P a ( L ) , ⊔ , ⊓ ) is depicted in Figure 5. ✉✉ ✉ ✉✉ ✉ ✉ ✉✉ ✉ ✉ ✉ ✉✉ ✉ ✉ ✉✉ ✉ ✉✉ ❅❅❅❅❅❅❅❅ (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)(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) ❅❅❅❅❅❅❅❅ (cid:0)(cid:0)(cid:0)(cid:0)❅❅❅❅❅❅❅❅(cid:0)(cid:0)(cid:0)(cid:0) ❅❅❅❅❅❅❅❅(cid:0)(cid:0)(cid:0)(cid:0) (0 , , d ) ( a,
1) (0 , c )( b, d ) ( a, d ) (0 , a ) ( a, c )( c, d ) ( b, a ) ( a, a ) ( a, b ) ( d, c )( c, a ) ( a,
0) ( d, a ) ( d, b )( c,
0) (1 , a ) ( d, , Define an antitone involution ′ on ( L, ≤ ) and binary operations · and → on L by x a b c d x ′ c d a b and x · y = (cid:26) x ∧ y ∧ b if x, y ∈ { a, c } ,x ∧ y otherwise x → y = (cid:26) x ′ ∨ y ∨ d if x, y ∈ { a, c } ,x ′ ∨ y otherwisefor all x, y ∈ L . Then ( L, ∨ , ∧ , · , → , is an integral commutative residuated lattice and x ′ = x → for all x ∈ L . Hence there holds the double negation law. Since a is neither dempotent with respect to · nor meet-irreducible nor comparable with all elements of L ,we cannot apply Theorem 4.1. However, since L is distributive, a is atom of L andconditions (i) and (ii) of Corollary 4.5 are satisfied, P a ( L ) is closed with respect to ⊙ andhence also with respect to ⇒ . If L denotes the lattice from Example 4.7 then ( c, d ) , ( d, b ) ∈ P d ( L ), but( c, d ) ⊙ ( d, b ) = ( c · d, ( c → b ) ∧ ( d → d )) = ( c ∧ d, ( c ′ ∨ b ) ∧ ( d ′ ∨ d )) == ( a, ( a ∨ b ) ∧ ( b ∨ d )) = ( a, c ∧
1) = ( a, c ) / ∈ P d ( L )since d c = a ∨ c . This shows that P d ( L ) is not closed with respect to ⊙ (and hencealso not with respect to ⇒ ).If L satisfies the double negation law then, because of (vi) of Proposition 2.2, ′ is anantitone involution on ( L, ≤ ). Two elements a and b of L are said to be orthogonal toeach other (shortly, a ⊥ b ) if a ≤ b ′ . If L satisfies the double negation law then this isequivalent to b ≤ a ′ . ( L , ⊔ , ⊓ , ⊙ , ⇒ , (0 , , (1 , double negationlaw for orthogonal elements if ( x, y ) ′′ = ( x, y ) for all ( x, y ) ∈ L with x ⊥ y where( x, y ) ′ := ( x, y ) ⇒ (0 ,
1) for all ( x, y ) ∈ L . Theorem 4.8.
Let L = ( L, ∨ , ∧ , · , → , , be a bounded commutative residuated latticesatisfying the double negation law and a ∈ L . Then the full twist-product ( L , ⊔ , ⊓ , ⊙ , ⇒ , (0 , , (1 , is a commutative residuated lattice with zero-element (0 , satisfying thedouble negation law for orthogonal elements.Proof. If a, b ∈ L and a ⊥ b then b ≤ a ′ and hence( a, b ) ′ = ( a, b ) ⇒ (0 ,
1) = (( a → ∧ (1 → b ) , a ·
1) = ( a ′ ∧ b, a ) = ( b, a ) , ( a, b ) ′′ = ( b, a ) ′ = ( a, b ) . In this section we show that the operations ⊙ and ⇒ on the full twist-product ( L , ⊔ , ⊓ )can be defined also in a way different from (1) and (2) such that ( L , ⊔ , ⊓ , ⊙ , ⇒ , (0 , , (1 , Theorem 5.1.
Let ( L, ∨ , ∧ , · , → , , be a bounded commutative residuated lattice satis-fying the double negation law and define x ′ := x → for all x ∈ L and ( x, y ) ⊙ ( z, v ) := ( x · z, ( y ′ · v ′ ) ′ ) , (5)( x, y ) ⇒ ( z, v ) := ( x → z, ( y ′ → v ′ ) ′ ) (6) for all ( x, y ) , ( z, v ) ∈ L . Then ( L , ⊔ , ⊓ , ⊙ , ⇒ , (0 , , (1 , is a bounded commutativeresiduated lattice satisfying the double negation law. roof. Let a, b, c, d, e, f ∈ L . Obviously, ( L , ⊔ , ⊓ , (0 , , (1 , x, y ) ⊙ ( z, v ) ≈ ( x · z, ( y ′ · v ′ ) ′ ) ≈ ( z · x, ( v ′ · y ′ ) ′ ) ≈ ( z, v ) ⊙ ( x, y ) , (( x, y ) ⊙ ( z, v )) ⊙ ( t, w ) ≈ ( x · z, ( y ′ · v ′ ) ′ ) ⊙ ( t, w ) ≈ (( x · z ) · t, (( y ′ · v ′ ) · w ′ ) ′ ) ≈≈ ( x · ( z · t ) , ( y ′ · ( v ′ · w ′ )) ′ ) ≈ ( x, y ) ⊙ ( z · t, ( v ′ · w ′ ) ′ ) ≈≈ ( x, y ) ⊙ (( z, v ) ⊙ ( t, w )) , ( x, y ) ⊙ (1 , ≈ ( x · , ( y ′ · ′ ) ′ ) ≈ ( x, ( y ′ · ′ ) ≈ ( x, y ′′ ) ≈ ( x, y ) . Moreover, the following are equivalent:( a, b ) ⊙ ( c, d ) ≤ ( e, f ) , ( a · c, ( b ′ · d ′ ) ′ ) ≤ ( e, f ) ,a · c ≤ e and f ≤ ( b ′ · d ′ ) ′ ,a · c ≤ e and b ′ · d ′ ≤ f ′ ,a · c ≤ e and b ′ ≤ d ′ → f ′ ,a ≤ c → e and ( d ′ → f ′ ) ′ ≤ b, ( a, b ) ≤ ( c → e, ( d ′ → f ′ ) ′ ) , ( a, b ) ≤ ( c, d ) ⇒ ( e, f ) . Finally, we have( x, y ) ′ ≈ ( x, y ) ⇒ (0 , ≈ ( x → , ( y ′ → ′ ) ′ ) ≈ ( x ′ , ( y ′ → ′ ) ≈ ( x ′ , y ′′′ ) ≈ ( x ′ , y ′ ) , ( x, y ) ′′ ≈ ( x ′ , y ′ ) ′ ≈ ( x ′′ , y ′′ ) ≈ ( x, y ) . Remark 5.2.
Let us note that under the assumptions of Theorem 5.1, the antitoneinvolution ( x, y ) ′ := ( x ′ , y ′ ) in the full twist-product L as well as in P a ( L ) can be derivedin a natural way by ( x, y ) ′ ≈ ( x, y ) ⇒ (0 , since ( x, y ) ⇒ (0 , ≈ ( x → , ( y ′ → ′ ) ′ ) ≈ ( x ′ , ( y ′ → ′ ) ≈ ( x ′ , y ′′′ ) ≈ ( x ′ , y ′ ) . This does not hold if ⊙ and ⇒ are defined by (1) and (2) , respectively. Remark 5.3.
It is worth noticing that the case when the operations ⊙ and ⇒ are de-fined by (5) and (6) , respectively, has an interpretation e.g. in MV -algebras. Namely, an MV -algebra is an algebra ( M, ⊕ , ¬ , of type (2 , , where ( M, ⊕ , is a commutativemonoid, ¬ satisfies the identity ¬¬ x ≈ x and ⊕ and ¬ are related by the Lukasiewiczaxiom ¬ ( ¬ x ⊕ y ) ⊕ y ≈ ¬ ( ¬ y ⊕ x ) ⊕ x. Then ( M, ∨ , ∧ ) becomes a distributive lattice where x ∨ y := ¬ ( ¬ x ⊕ y ) ⊕ y,x ∧ y := ¬ ( ¬ x ∨ ¬ y ) . for all x, y ∈ M . MV -algebras serve as an algebraic semantics of the many-valued Lukasiewicz logics, ⊕ is interpreted as disjunction and → defined by x → y := ¬ x ⊕ y or all x, y ∈ M as implication. If we put x · y := ¬ ( ¬ x ⊕ ¬ y ) for all x, y ∈ M then x → y ≈ ¬ ( x · ¬ y ) and ( M, ∨ , ∧ , · , → , , forms a bounded residuated lattice satisfyingthe double negation law. If we now define ⊙ and ⇒ on the full twist-product M by (5) and (6) , respectively, we obtain ( x, y ) ⊙ ( z, v ) ≈ ( x · z, y ⊕ v ) , ( x, y ) ⇒ ( z, v ) ≈ ( x → z, ¬ y · v ) . In fact, the lattice L = ( L, ∨ , ∧ ) from Example 4.7 is an MV -algebra where ¬ x := x → and x ⊕ y := ¬ ( ¬ x · ¬ y ) for all x, y ∈ L . It was shown in [3] for Kleene lattices and in [4] for pseudo-Kleene lattices ( L ∨ , ∧ , ′ ) thatthere exists at most one element a of L satisfying a ′ = a . If such an element exists in alattice with an antitone involution, we can prove the following result. Theorem 5.4.
Let L = ( L, ∨ , ∧ , ′ ) be a lattice with an antitone involution and a ∈ L with a ′ = a , assume that ( P a ( L ) , ⊔ , ⊓ ) is a sublattice of ( L , ⊔ , ⊓ ) and ( x, y ) ′ = ( x ′ , y ′ ) for all ( x, y ) ∈ L . Then ( P a ( L ) , ⊔ , ⊓ , ′ ) is a pseudo-Kleene lattice if and only if L hasthis property.Proof. Let b, c ∈ L and ( d, e ) , ( f, g ) ∈ P a ( L ). We have ( x, y ) ′ ≈ ( x, y ) ⇒ (0 , ≈ ( x ′ , y ′ ) as explained in Remark 5.2. If ( P a ( L ) , ⊔ , ⊓ , ′ ) is a pseudo-Kleene lattice then( b, a ) , ( c, a ) ∈ P a ( L ) and hence( b ∧ b ′ , a ∨ a ′ ) = ( b, a ) ⊓ ( b ′ , a ′ ) = ( b, a ) ⊓ ( b, a ) ′ ≤ ( c, a ) ⊔ ( c, a ) ′ = ( c, a ) ⊔ ( c ′ , a ′ ) = ( c ∨ c ′ , a ∧ a ′ ) , i.e. b ∧ b ′ ≤ c ∨ c ′ showing that L is a pseudo-Kleene lattice. Conversely, assume L tobe a pseudo-Kleene lattice. Then d ∧ e ≤ a ≤ d ∨ e whence d ′ ∧ e ′ ≤ a ′ ≤ d ′ ∨ e ′ , i.e. d ′ ∧ e ′ ≤ a ≤ d ′ ∨ e ′ which shows ( d, e ) ′ = ( d ′ , e ′ ) ∈ P a ( L ). Hence P a ( L ) is closed withrespect to ′ . Finally, we have( d, e ) ⊓ ( d, e ) ′ = ( d, e ) ⊓ ( d ′ , e ′ ) = ( d ∧ d ′ , e ∨ e ′ ) ≤ ( f ∨ f ′ , g ∧ g ′ ) = ( f, g ) ⊔ ( f ′ , g ′ ) == ( f, g ) ⊔ ( f, g ) ′ showing that ( P a ( L ) , ⊔ , ⊓ , ′ ) is a pseudo-Kleene lattice.Our next aim is to show when P a ( L ) is closed under the operation ⊙ defined by (5). Weprove the following. Theorem 5.5.
Let L = ( L, ∨ , ∧ , · , → , ′ , be a commutative residuated lattice with an an-titone involution, let a ∈ L be idempotent with respect to · , ∨ -irreducible and ∧ -irreducible,assume a ′ · a ′ = a ′ and define ⊙ by (5) . Then P a ( L ) is closed with respect to ⊙ .Proof. Let ( b, c ) , ( d, e ) ∈ P a ( L ). We have( x, y ) ⊙ ( z, v ) ≈ ( x · z, ( y ′ · v ′ ) ′ ) ≈ ( z · x, ( v ′ · y ′ ) ′ ) ≈ ( z, v ) ⊙ ( x, y ) . According to Theorem 3.6, P a ( L ) = { ( x, y ) ∈ L | ( x, y ) is comparable with ( a, a ) } . In the following we often use (i) and (ii) of Proposition 2.2.16i) Assume ( b, c ) ≤ ( a, a ).We have b · d ≤ b ≤ a and every one of the following statements implies the nextone: a ≤ c,c ′ ≤ a ′ ,c ′ · e ′ ≤ a ′ ,a ≤ ( c ′ · e ′ ) ′ . This shows ( b, c ) ⊙ ( d, e ) = ( b · d, ( c ′ · e ′ ) ′ ) ≤ ( a, a ).(ii) Assume ( d, e ) ≤ ( a, a ).Then ( b, c ) ⊙ ( d, e ) = ( d, e ) ⊙ ( b, c ) ≤ ( a, a ).(iii) Assume ( a, a ) ≤ ( b, c ) , ( d, e ).Then c, e ≤ a and hence a ′ ≤ c ′ , e ′ whence a ′ = a ′ · a ′ ≤ c ′ · a ′ ≤ c ′ · e ′ from which weconclude ( c ′ · e ′ ) ′ ≤ a . Because of a ≤ b, d we have a = a · a ≤ b · a ≤ b · d . Togetherwe obtain ( a, a ) ≤ ( b · d, ( c ′ · e ′ ) ′ ) = ( b, c ) ⊙ ( d, e ).Unfortunately, P a ( L ) is not closed under ⇒ defined by (C2) provided L in non-trivial,i.e. if it has more than one element. Theorem 5.6.
Let ( L, ∨ , ∧ , · , → , ′ , , be a bounded commutative residuated lattice withan antitone involution and a ∈ L and put ( x, y ) ⇒ ( z, v ) := ( x → z, ( y ′ → v ′ ) ′ ) for all ( x, y ) , ( z, v ) ∈ L . Then P a ( L ) is closed with respect to ⇒ if and only if | L | = 1 .Proof. Assume P a ( L ) to be closed with respect to ⇒ . Since (0 , a ) , ( a, , ( a, , (1 , a ) ∈ P a ( L ) we have(0 ,
0) = (0 , ′ ) = (1 → , ( a ′ → a ′ ) ′ ) = (1 , a ) ⇒ (0 , a ) ∈ P a ( L ) , (1 ,
1) = (1 , ′ ) = (1 , (1 → ′ ) = ( a → a, (0 ′ → ′ ) ′ ) = ( a, ⇒ ( a, ∈ P a ( L )whence a ≤ ∨ ≤ ∧ ≤ a and therefore 0 = 1, i.e. | L | = 1. References [1] R. Bˇelohl´avek, Fuzzy Relational Systems. Foundations and Principles. Springer, newYork 2002. ISBN 978-1-4613-5168-9.[2] M. Busaniche and R. Cignoli, The subvariety of commutative residuated lattices rep-resented by twist-products. Algebra Universalis (2014), 5–22.[3] I. Chajda, A note on pseudo-Kleene algebras, Acta Univ. Palack. Olomuc. Fac. RerumNatur., Math. (2016), 39–45.[4] I. Chajda and H. L¨anger, Kleene posets and pseudo-Kleene posets. Miskolc Math.Notes (submitted). http://arxiv.org/abs/2006.04417.175] R. Cignoli, Injective De Morgan and Kleene algebras. Proc. Amer. Math. Soc. (1975), 269–278.[6] J. A. Kalman, Lattices with involution. Trans. Amer. Math. Soc. (1958), 485–491.[7] C. Tsinakis and A. M. Wille, Minimal varieties of involutive residuated lattices. StudiaLogica83