Filters and congruences in sectionally pseudocomplemented lattices and posets
aa r X i v : . [ m a t h . L O ] J u l Filters and congruences in sectionallypseudocomplemented lattices and posets
Ivan Chajda and Helmut L¨anger
Abstract
In our previous papers, together with J. Paseka we introduced so-called section-ally pseudocomplemented lattices and posets and illuminated their role in algebraicconstructions. We believe that – similar to relatively pseudocomplemented lattices– these structures can serve as an algebraic semantics of certain intuitionistic log-ics. The aim of the present paper is to define congruences and filters in thesestructures, derive mutual relationships between them and describe basic propertiesof congruences in strongly sectionally pseudocomplemented posets. For the descrip-tion of filters both in sectionally pseudocomplemented lattices and posets, we usethe tools introduced by A. Ursini, i.e. ideal terms and the closedness with respectto them. It seems to be of some interest that a similar machinery can be appliedalso for strongly sectionally pseudocomplemented posets in spite of the fact thatthe corresponding ideal terms are not everywhere defined.
AMS Subject Classification:
Keywords:
Sectionally pseudocomplemented lattice, sectionally pseudocomplementedposet, filter, congruence, weak regularity, congruence permutability, Maltsev term, idealterm, closedness of a subset, congruence class, deductive system, partial term
The concept of a relative pseudocomplemented lattice was introduced by R. P. Dilworth([5]). It was used in several branches of mathematics, e.g. as an algebraic axiomatizationof intuitionistic logic (by Heyting and Brouwer) where the relative pseudocomplement isinterpreted as the logical connective implication.However, every relative pseudocomplemented lattice is distributive, see e.g. [1] and [6].Because not every non-classical propositional calculus is necessarily distributive (for in-stance, the logic of quantum mechanics), it was a question if the concept of relativepseudocomplementation can be extended in a reasonable way to non-distributive lattices. Support of the research of the authors by the Austrian Science Fund (FWF), project I 4579-N, and theCzech Science Foundation (GA ˇCR), project 20-09869L, entitled “The many facets of orthomodularity”,as well as by ¨OAD, project CZ 02/2019, entitled “Function algebras and ordered structures related tologic and data fusion”, and, concerning the first author, by IGA, project PˇrF 2020 014, is gratefullyacknowledged.
Recall that a lattice ( L, ∨ , ∧ ) is said to be sectionally pseudocomplemented if for all a, b ∈ L there exists the pseudocomplement of a ∨ b in the interval ([ b ) , ≤ ), i.e. the greatest element c of L satisfying ( a ∨ b ) ∧ c = b. In this case c is called the sectional pseudocomplement of a with respect to b and it willbe denoted by a ∗ b . We consider sectionally pseudocomplemented lattices as algebras( L, ∨ , ∧ , ∗ ) of type (2 , , x ∗ x . In the following we consideronly non-empty lattices.An example of a sectionally pseudocomplemented lattice that is not relatively pseudo-complemented is N depicted in Figure 1: ✉ ✉✉ ✉✉ ❇❇❇❇❇❇ ✁✁✁✁✂✂✂✂✂✂ ❆❆❆❆ acb ∗ a b c
10 1 1 1 1 1 a b b b c a c c b a b a b c Proposition 2.1.
The class of sectionally pseudocomplemented lattices ( L, ∨ , ∧ , ∗ ) formsa variety which besides the lattice axioms is determined by the following identities: z ∨ y ≤ x ∗ (( x ∨ y ) ∧ ( z ∨ y )) , ( x ∨ y ) ∧ ( x ∗ y ) ≈ y. This variety is congruence permutable, congruence distributive and weakly regular. AMaltsev term for congruence permutability is given by p ( x, y, z ) := (( x ∗ y ) ∗ z ) ∧ (( z ∗ y ) ∗ x ) . For the concept of congruence permutability we refer the reader to [3].Weak regularity means that every congruence Θ on a sectionally pseudocomplementedlattice with greatest element 1 is determined by its kernel, i.e. by the congruence class[1]Θ. Hence our first task is to describe these classes. For this purpose we introduce thefollowing concept:
Definition 2.2.
Let L = ( L, ∨ , ∧ , ∗ ) be a sectionally pseudocomplemented lattice. A filter of L is a subset F of L containing such that x ∗ y, y ∗ x ∈ F implies ( x ∨ z ) ∗ ( y ∨ z ) , ( x ∧ z ) ∗ ( y ∧ z ) , ( x ∗ z ) ∗ ( y ∗ z ) , ( z ∗ x ) ∗ ( z ∗ y ) ∈ F. Let
Fil L denote the set of all filters of L . For any subset M of L define a binary relation Φ( M ) on L as follows: Φ( M ) := { ( x, y ) ∈ L | x ∗ y, y ∗ x ∈ M } . The following results were proved in [2] and [4].
Lemma 2.3. If L = ( L, ∨ , ∧ , ∗ ) is a sectionally pseudocomplemented lattice and a, b, c ∈ L then (i) a ∗ b = 1 if and only if a ≤ b , (ii) 1 ∗ a = a , (iii) a ≤ b ∗ a , (iv) a ≤ ( a ∗ b ) ∗ b , (v) if a ≤ b then b ∗ c ≤ a ∗ c , (vi) ( a ∨ b ) ∧ ( a ∗ b ) = b . Observe that (iii) implies b ≤ ( a ∗ b ) ∗ b .The relationship between congruences and filters in sectionally pseudocomplemented lat-tices is illuminated in the next two theorems.3 heorem 2.4. Let L = ( L, ∨ , ∧ , ∗ ) be a sectionally pseudocomplemented lattice and Θ ∈ Con L . Then [1]Θ ∈ Fil L and for any x, y ∈ L , ( x, y ) ∈ Θ if and only if x ∗ y, y ∗ x ∈ [1]Θ , i.e. Φ([1]Θ) = Θ .Proof.
Let a, b ∈ L . If ( a, b ) ∈ Θ then a ∗ b, b ∗ a ∈ [ a ∗ a ]Θ = [1]Θ, i.e. ( a, b ) ∈ Φ([1]Θ).Conversely, if ( a, b ) ∈ Φ([1]Θ) then a ∗ b, b ∗ a ∈ [1]Θ and hence, using (ii) and (iv) ofLemma 2.3, a = a ∧ (( a ∗ b ) ∗ b ) Θ (1 ∗ a ) ∧ (1 ∗ b ) Θ (( b ∗ a ) ∗ a ) ∧ b = b, i.e. ( a, b ) ∈ Θ. This shows Φ([1]Θ) = Θ. Due to the substitution property of Θ withrespect to ∨ , ∧ and ∗ we see that [1]Θ satisfies the conditions from Definition 2.2 andhence [1]Θ ∈ Fil L .Theorem 2.4 witnesses that sectionally pseudocomplemented lattices are weakly regular.We can prove also the converse. Theorem 2.5.
Let L = ( L, ∨ , ∧ , ∗ ) be a sectionally pseudocomplemented lattice and F ∈ Fil L . Then Φ( F ) ∈ Con L and [1](Φ( F )) = F .Proof. Let a, b, c ∈ L . Evidently, Φ( F ) is symmetric and since 1 ∈ F and x ∗ x ≈ a ∗ b, b ∗ a ∈ F . Then by Definition 2.2( a ∗ c ) ∗ ( b ∗ c ) , ( b ∗ c ) ∗ ( a ∗ c ) , ( c ∗ a ) ∗ ( c ∗ b ) , ( c ∗ b ) ∗ ( c ∗ a ) , ( a ∨ c ) ∗ ( b ∨ c ) , ( b ∨ c ) ∗ ( a ∨ c ) , ( a ∧ c ) ∗ ( b ∧ c ) , ( b ∧ c ) ∗ ( a ∧ c ) ∈ F whence ( a ∗ c, b ∗ c ) , ( c ∗ a, c ∗ b ) , ( a ∨ c, b ∨ c ) , ( a ∧ c, b ∧ c ) ∈ Φ( F ) . Hence Φ( F ) has the substitution property with respect to all basic operations of L . Sincethe variety of sectionally pseudocomplemented lattices is congruence permutable, Φ( F )is also transitive, see e.g. Werner’s Theorem ([8]) or Corollary 3.1.13 in [3], and henceΦ( F ) ∈ Con L . Finally, the following are equivalent: a ∈ [1](Φ( F )) , ( a, ∈ Φ( F ) ,a ∗ , ∗ a ∈ F, , a ∈ F,a ∈ F and hence [1](Φ( F )) = F .It is elementary to check that for every sectionally pseudocomplemented lattice L ,(Fil L , ⊆ ) is a complete lattice. 4 xample 2.6. The sectionally pseudocomplemnted lattice from Fig. 1 has the followingfilters: F (1) = { } ,F ( a ) = F ( c ) = { a, c, } ,F (0) = F ( b ) = { , a, b, c, } . The following corollary follows from Theorems 2.4 and 2.5.
Corollary 2.7.
For every sectionally pseudocomplemented lattice L the mappings Φ [1]Φ and F Φ( F ) are mutually inverse isomorphisms between the complete lattices (Con L , ⊆ ) and (Fil L , ⊆ ) . Let ( L, ∨ , ∧ , ∗ ) be a sectionally pseudocomplemented lattice. A deductive system of L isa subset D of L containing 1 and satisfying the following condition:If a ∈ D, b ∈ L and a ∗ b ∈ D then b ∈ D. In the following ( F ∗ ( F ∗ a )) ∗ a denotes the set { ( x ∗ ( y ∗ a )) ∗ a | x, y ∈ F } . Analogously,we proceed in similar cases. Theorem 2.8.
Let L = ( L, ∨ , ∧ , ∗ ) be a sectionally pseudocomplemented lattice, Θ ∈ Con L , F ∈ Fil L and a, b ∈ L . Then (i) Every class of Θ is a convex subset of ( L, ≤ ) , (ii) F is a deductive system of L , (iii) F is a lattice filter of L , (iv) a ∗ ( F ∧ a ) ⊆ F and ( F ∗ ( F ∗ a )) ∗ a ⊆ F .Proof. (i) If c, d ∈ [ a ]Θ and c ≤ b ≤ d then b = c ∨ b ∈ [ d ∨ b ]Θ = [ d ]Θ = [ a ]Θ . (ii) If a, a ∗ b ∈ F then b = 1 ∗ b ∈ [ a ∗ b ](Φ( F )) = [1](Φ( F )) = F. (iii) If a ∈ F then a ∨ b ∈ [1 ∨ b ](Φ( F )) = [1](Φ( F )) = F. Moreover, if a, b ∈ F then a ∧ b ∈ [1 ∧ F )) = [1](Φ( F )) = F. (iv) a ∗ ( F ∧ a ) ⊆ [ a ∗ (1 ∧ a )](Φ( F )) = [ a ∗ a ](Φ( F )) = [1](Φ( F )) = F, ( F ∗ ( F ∗ a )) ∗ a ⊆ [(1 ∗ (1 ∗ a )) ∗ a ](Φ( F )) = [(1 ∗ a ) ∗ a ](Φ( F )) = [ a ∗ a ](Φ( F )) == [1](Φ( F )) = F. Sectionally pseudocomplemented posets
Now we turn our attention to sectionally pseudocomplemented posets.
Definition 3.1.
Let P = ( P, ≤ ) be a poset. Then P is called sectionally pseudocomple-mented if for all a, b ∈ P there exists a greatest element c of P satisfying L ( U ( a, b ) , c ) = L ( b ) . This element c is called the sectional pseudocomplement a ∗ b of a with respect to b . Wewrite sectionally pseudocomplemented posets in the form ( P, ≤ , ∗ ) . A strongly sectionallypseudocomplemented poset is an ordered quadruple ( P, ≤ , ∗ , such that ( P, ≤ , ∗ ) is asectionally pseudocomplemented poset with greatest element satisfying the identity x ≤ ( x ∗ y ) ∗ y. The following results were proved in [4].
Lemma 3.2. If P = ( P, ≤ , ∗ ) is a sectionally pseudocomplemented poset with greatestelement and a, b, c ∈ P then (i) a ∗ b = 1 if and only if a ≤ b , (ii) 1 ∗ a = a , (iii) a ≤ b ∗ a , (iv) if b ≤ a then a ≤ ( a ∗ b ) ∗ b , (v) if a ≤ b then b ∗ c ≤ a ∗ c , (vi) L ( U ( a, b ) , a ∗ b ) = L ( b ) . Observe that (iii) implies b ≤ ( a ∗ b ) ∗ b . Hence in case a ≤ b we have a ≤ ( a ∗ b ) ∗ b .It is easy to see that every sectionally pseudocomplemented lattice is a strongly sectionallypseudocomplemented poset, and a lattice is sectionally pseudocomplemented if and onlyif it is sectionally pseudocomplemented as a poset. Remark 3.3. If ( P, ≤ , ∗ ) is a sectionally pseudocomplemented poset and a, b ∈ P then L ( U ( a, b ) , a ∗ b ) = L ( b ) which shows that there exists the infimum U ( a, b ) ∧ ( a ∗ b ) and hence the previous isequivalent to U ( a, b ) ∧ ( a ∗ b ) = b. Thus, in case a ≥ b we obtain a ∧ ( a ∗ b ) = b . An example of a strongly sectionally pseudocomplemented poset which is not a lattice isvisualized in Figure 2. 6 ✉ ✉✉✉ ✉✉ ❆❆❆❆ ✁✁✁✁❆❆❆❆❆❆❆❆(cid:0)(cid:0)(cid:0)(cid:0)✁✁✁✁ ❆❆❆❆ a bcd e ∗ is as follows: ∗ a b c d e
10 1 1 1 1 1 1 1 a b b b c a c c b a b d a b c e e a b c d a b c d e c with respect to a does not exist.It should be noted that there are sectionally pseudocomplemented posets which are notstrongly sectionally pseudocomplemented, see e.g. [4], but these are rather curious.Since a sectionally pseudocomplemented poset P has only one operation, namely ∗ , a con-gruence on P should satisfy the substitution property with respect to ∗ . However, thiscondition is rather weak and we cannot expect to obtain a natural relationship betweencongruences and congruence kernels similar to that obtained for sectionally pseudocom-plemented lattices in the previous section. Namely, our concept of a congruence on astrongly sectionally pseudocomplemented poset should respect also some aspects of thepartial order relation. This is the reason why we introduce the following property. Definition 3.4. A binary relation ρ on a poset is called min-stable if the following holds:If ( a, b ) , ( c, d ) ∈ ρ , a is comparable with c and b is comparable with d then (min( a, c ) , min( b, d )) ∈ ρ. Observe that this condition trivially holds if a ≤ c and b ≤ d or if a ≥ c and b ≥ d .Now we can define Definition 3.5.
Let P = ( P, ≤ , ∗ ) be a sectionally pseudocomplemented poset. A congru-ence on P is a min -stable congruence on the algebraic reduct ( P, ∗ , of P . Let Con P denote the set of all congruences on P . L may not co-incide with the congruences on L if it is considered only as a sectionally pseudocomple-mented poset.In analogy to the lattice case we define Definition 3.6.
Let P = ( P, ≤ , ∗ , be a sectionally pseudocomplemented poset withgreatest element . A filter of P is a subset F of P containing and satisfying thefollowing conditions for all x, y, z, v ∈ P : • If x ∗ y, y ∗ x ∈ F then ( x ∗ z ) ∗ ( y ∗ z ) , ( z ∗ x ) ∗ ( z ∗ y ) ∈ F , • if x ∗ y, y ∗ x, z ∗ v, v ∗ z ∈ F , x and z are comparable and y and v are comparablethen min( x, z ) ∗ min( y, v ) ∈ F .Let Fil P denote the set of all filters of P . It is elementary to check that for every stronglysectionally pseudocomplemented poset P , (Con P , ⊆ ) and (Fil P , ⊆ ) are complete lattices.For any subset M of P put Φ( M ) := { ( x, y ) ∈ P | x ∗ y, y ∗ x ∈ M } . The relationship between congruences and filters in strongly sectionally pseudocomple-mented posets is illuminated in the next two theorems.
Theorem 3.7.
Let P = ( P, ≤ , ∗ , be a strongly sectionally pseudocomplemented posetand Θ ∈ Con L . Then [1]Θ ∈ Fil L and for any x, y ∈ P , ( x, y ) ∈ Θ if and only if x ∗ y, y ∗ x ∈ [1]Θ , i.e. Φ([1]Θ) = Θ .Proof.
Let a, b ∈ L . If ( a, b ) ∈ Θ then, by Lemma 3.2, a ∗ b, b ∗ a ∈ [ a ∗ a ]Θ = [1]Θ, i.e.( a, b ) ∈ Φ([1]Θ). Conversely, if ( a, b ) ∈ Φ([1]Θ) then a ∗ b, b ∗ a ∈ [1]Θ and hence, usingagain Lemma 3.2, ( a, ( b ∗ a ) ∗ a ) = (1 ∗ a, ( b ∗ a ) ∗ a ) ∈ Θ , (( a ∗ b ) ∗ b, b ) = (( a ∗ b ) ∗ b, ∗ b ) ∈ Θ . Since P is strongly sectionally pseudocomplemented we have a ≤ ( a ∗ b ) ∗ b and ( b ∗ a ) ∗ a ≥ b ,thus by min-stability of Θ we conclude( a, b ) = (min( a, ( a ∗ b ) ∗ b ) , min(( b ∗ a ) ∗ a, b )) ∈ Θ . This shows Φ([1]Θ) = Θ. Due to the substitution property of Θ with respect to ∗ andthe min-stability of Θ we obtain [1]Θ ∈ Fil L .We have shown that every congruence Θ on a strongly sectionally pseudocomplementedposet is fully determined by its 1-class [1]Θ. Hence we conclude Corollary 3.8.
Strongly sectionally pseudocomplemented posets are weakly regular.
We can prove also the converse. 8 heorem 3.9.
Let P = ( P, ≤ , ∗ , be a strongly sectionally pseudocomplemented posetand F ∈ Fil P . Then Φ( F ) ∈ Con P and [1](Φ( F )) = F .Proof. Let a, b, c, d ∈ P . Evidently, Φ( F ) is symmetric and since 1 ∈ F and x ∗ x ≈ a, b ) ∈ Φ( F ) then a ∗ b, b ∗ a ∈ F and hence, using the propertieslisted in Definition 3.6, ( a ∗ c ) ∗ ( b ∗ c ) , ( b ∗ c ) ∗ ( a ∗ c ) ∈ F, ( c ∗ a ) ∗ ( c ∗ b ) , ( c ∗ b ) ∗ ( c ∗ a ) ∈ F. Thus ( a ∗ c, b ∗ c ) , ( c ∗ a, c ∗ b ) ∈ Φ( F ). Hence Φ( F ) has the substitution property withrespect to ∗ . Moreover, if ( a, b ) , ( c, d ) ∈ Φ( F ) then a ∗ b, b ∗ a, c ∗ d, d ∗ c ∈ F and byDefinition 3.6 min( a, c ) ∗ min( b, d ) , min( b, d ) ∗ min( a, c ) ∈ F, i.e. (min( a, c ) , min( b, d )) ∈ Φ( F ). This shows that Φ( F ) is min-stable. If ( a, b ) , ( b, c ) ∈ Φ( F ) then ( a ∗ b ) ∗ b Φ( F ) ( b ∗ b ) ∗ c = 1 ∗ c = c,a = 1 ∗ a = ( b ∗ b ) ∗ a Φ( F ) ( c ∗ b ) ∗ b and hence using min-stability of Φ( F )( a, c ) = (min(( a ∗ b ) ∗ b, a ) , min( c, ( c ∗ b ) ∗ b )) ∈ Φ( F ) , i.e. Φ( F ) is transitive. Therefore Φ( F ) ∈ Con P . Finally, the following are equivalent: a ∈ [1](Φ( F )) , ( a, ∈ Φ( F ) ,a ∗ , ∗ a ∈ F, , a ∈ F,a ∈ F. This shows [1](Φ( F )) = F . Example 3.10.
The lattice of filters of the strongly sectionally pseudocomplemnted posetfrom Figure 2 consists of the following six filters: F (1) = { } ,F ( d ) = { d, } ,F ( e ) = { e, } ,F ( { d, e } ) = { d, e, } ,F ( a ) = F ( c ) = { a, c, d, e, } ,F (0) = F ( b ) = { , a, b, c, d, e, } . The corresponding Hasse diagram is depicted in Figure 3: ✉ ✉✉✉✉ ❆❆❆❆ ✁✁✁✁✁✁✁✁ ❆❆❆❆ F (1) F ( d ) F ( e ) F ( { d, e } ) F ( a ) F (0)Fig. 3The following corollary follows from Theorems 3.7 and 3.9. Corollary 3.11.
For every strongly sectionally pseudocomplemented poset P the map-pings Φ [1]Φ and F Φ( F ) are mutually inverse isomorphisms between the completelattices (Con P , ⊆ ) and (Fil P , ⊆ ) . Using the min-stability property of congruences in strongly sectionally pseudocomple-mented posets we can prove
Theorem 4.1.
Let P = ( P, ≤ , ∗ , be a strongly sectionally pseudocomplemented posetand Θ ∈ Con P . Then every class of Θ is a convex subset of ( P, ≤ ) .Proof. If a, c ∈ P , b, d ∈ [ a ]Θ and b ≤ c ≤ d then( c ∗ d ) ∗ b = 1 ∗ b = b ≤ c ≤ ( c ∗ b ) ∗ b, (( c ∗ d ) ∗ b, ( c ∗ b ) ∗ b ) ∈ Θand hence by min-stability of Θ we obtain( b, c ) = (min(( c ∗ d ) ∗ b, c ) , min(( c ∗ b ) ∗ b, c )) ∈ Θ , which implies c ∈ [ b ]Θ = [ a ]Θ.We now investigate quotients P / Θ of strongly sectionally pseudocomplemented posets P with respect to its congruences.Let P = ( P, ≤ , ∗ ,
1) be a strongly sectionally pseudocomplemented poset and Θ ∈ Con P .We define a binary relation ≤ ′ on P/ Θ byfor all a, b ∈ P, [ a ]Θ ≤ ′ [ b ]Θ if and only if [ a ]Θ ∗ [ b ]Θ = [1]Θ . poset ( P, ≤ ) is called up-directed if for any x, y ∈ P there exists some z ∈ P with x, y ≤ z . Hence, every poset having a greatest element is up-directed.It should be mentioned that the poset ( P/ Θ , ≤ ′ ) where P = ( P, ≤ , ∗ ,
1) denotes thestrongly sectionally pseudocomplemented poset from Figure 2 and Θ the congruence on P corresponding to the filter F ( { d, e } ) of P is isomorphic to the lattice from Figure 1.The following theorem was partly proved for congruences on the algebraic reduct ( P, ∗ )in [4]. Theorem 4.2.
Let P = ( P, ≤ , ∗ , be a strongly sectionally pseudocomplemented poset, n ≥ , a, a , . . . , a n , b ∈ P and Θ ∈ Con P . Then the following hold: (i) if a ≤ b then [ a ]Θ ≤ ′ [ b ]Θ , (ii) [ a ]Θ ≤ ′ [ b ]Θ if and only if there exists some c ∈ [ b ]Θ with a ≤ c , (iii) ( P/ Θ , ≤ ′ ) is a poset, (iv) Every class of Θ is up-directed, (v) U ([ a ]Θ , . . . , [ a n ]Θ) = { [ x ]Θ | x ∈ U ( a . . . . , a n ) } in ( P/ Θ , ≤ ′ ) .Proof. (i) If a ≤ b then a ∗ b = 1 whence a ∗ b Θ 1, i.e. [ a ]Θ ∗ [ b ]Θ = [ a ∗ b ]Θ = [1]Θ, thus[ a ]Θ ≤ ′ [ b ]Θ.(ii) If [ a ]Θ ≤ ′ [ b ]Θ then a ∗ b Θ 1 and hence a ≤ ( a ∗ b ) ∗ b ∈ [1 ∗ b ]Θ = [ b ]Θ. So onecan put c := ( a ∗ b ) ∗ b . If, conversely, there exists some c ∈ [ b ]Θ with a ≤ c thenaccording to (i) we have [ a ]Θ ≤ ′ [ c ]Θ = [ b ]Θ.(iii) Obviously, ≤ ′ is reflexive. Now assume [ a ]Θ ≤ ′ [ b ]Θ and [ b ]Θ ≤ ′ [ a ]Θ. Then, by (ii),there exists some c ∈ [ b ]Θ with a ≤ c . Because of [ c ]Θ = [ b ]Θ ≤ ′ [ a ]Θ there existssome d ∈ [ a ]Θ with c ≤ d . Since a ≤ c ≤ d , a, d ∈ [ a ]Θ and ([ a ]Θ , ≤ ′ ) is convex weconclude c ∈ [ a ]Θ. Therefore [ a ]Θ = [ c ]Θ = [ b ]Θ which proves antisymmetry of ≤ ′ .Finally, let c ∈ P and assume [ a ]Θ ≤ ′ [ b ]Θ and [ b ]Θ ≤ ′ [ c ]Θ. Then, by (ii) thereexists some e ∈ [ b ]Θ with a ≤ e and because of [ e ]Θ = [ b ]Θ ≤ ′ [ c ]Θ some f ∈ [ c ]Θwith e ≤ f . From a ≤ e ≤ f we have a ≤ f which implies [ a ]Θ ≤ ′ [ f ]Θ = [ c ]Θ by(i), proving transitivity of ≤ ′ .(iv) Let b, c ∈ [ a ]Θ. Then( b ∗ c ) ∗ c ∈ [( c ∗ c ) ∗ c ]Θ = [1 ∗ c ]Θ = [ c ]Θ = [ a ]Θ ,b ≤ ( b ∗ c ) ∗ c since P is strongly sectionally pseudocomplemented ,c ≤ ( b ∗ c ) ∗ c according to Lemma 3.2 (iii) . Thus ( b ∗ c ) ∗ c is a common upper bound of b and c within ([ a ]Θ , ≤ ).(v) Assume [ a ]Θ ∈ U ([ a ]Θ , . . . , [ a n ]Θ). According to (ii), for all i ∈ { , . . . , n } thereexists some b i ∈ [ a ]Θ with a i ≤ b i . Because of (iv), ([ a ]Θ , ≤ ′ ) is up-directed andhence there exists some c ∈ [ a ]Θ with b , . . . , b n ≤ c . This shows[ a ]Θ = [ c ]Θ ∈ { [ x ]Θ | x ∈ U ( a , . . . , a n ) } . The converse inclusion follows from (i).11rom (iv) we conclude that if ( P, ≤ ) satisfies the ascending chain condition (in particular,if P is finite) then every class of Θ has a greatest element.The following concept is inspired by the derivation rule Modus Ponens in the non-classicallogic based on a sectionally pseudocomplemented poset where ∗ models the logical con-nective implication.Let ( P, ≤ , ∗ ,
1) be a strongly sectionally pseudocomplemented poset. A deductive system of P is a subset D of P containing 1 and satisfying the following condition:If a ∈ D, b ∈ P and a ∗ b ∈ D then b ∈ D. We can prove the following result in analogy to the corresponding result for sectionallypseudocomplemented lattices.
Theorem 4.3.
Let P = ( P, ≤ , ∗ , be a strongly sectionally pseudocomplemented poset, F ∈ Fil P and c ∈ P . Then (i) F is a deductive system of P , (ii) F is an order filter of P , (iii) P ∗ F ⊆ F , (iv) c ∗ ( F ∧ c ) , ( F ∗ ( F ∗ c )) ∗ c ⊆ F .Proof. We use the fact that the filter F is the 1-class of the congruence Φ( F ).(i) If a ∈ F , b ∈ P and a ∗ b ∈ F then b = 1 ∗ b ∈ [ a ∗ b ](Φ( F )) = [1](Φ( F )) = F. (ii) If a ∈ F , b ∈ P and a ≤ b then a ∗ b = 1 ∈ F and hence b ∈ F by (i).(iii) If a ∈ P and b ∈ F then a ∗ b ∈ [ a ∗ F )) = [1](Φ( F )) = F .(iv) c ∗ ( F ∧ c ) ⊆ [ c ∗ (1 ∧ c )](Φ( F )) = [ c ∗ c ](Φ( F )) = [1](Φ( F )) = F, ( F ∗ ( F ∗ c )) ∗ c ⊆ [(1 ∗ (1 ∗ c )) ∗ c ](Φ( F )) = [(1 ∗ c ) ∗ c ](Φ( F )) = [ c ∗ c ](Φ( F )) == [1](Φ( F )) = F. Theorem 4.3 shows that every filter is a deductive system. However, our concept of afilter is rather complicated and it seems that not all the properties of a filter are necessaryto prove this assertion. We can prove
Proposition 4.4.
Let P = ( P, ≤ , ∗ , be a strongly sectionally pseudocomplementedposet and M a subset of P containing and satisfying ( M ∗ ( M ∗ x )) ∗ x ⊆ M for all x ∈ P . Then M is a deductive system of P . roof. Let a ∈ N and b ∈ P . We have 1 ∈ M . If a ≤ b then b = 1 ∗ b = ( a ∗ b ) ∗ b = ( a ∗ (1 ∗ b )) ∗ b ∈ ( N ∗ ( N ∗ b )) ∗ b ⊆ N. Hence, if a ∗ b ∈ N then because of a ≤ ( a ∗ b ) ∗ b we have ( a ∗ b ) ∗ b ∈ N which implies b = 1 ∗ b = ((( a ∗ b ) ∗ b )(( a ∗ b ) ∗ b )) ∗ b ∈ ( N ∗ ( N ∗ b )) ∗ b ⊆ N. Observe that the condition mentioned in Proposition 4.4 is just the second one of (iv) ofTheorem 4.3.For the concept of an ideal of a universal algebra which corresponds to our concept ofa filter and for the concept of ideal terms the reader is referred to [7]. In particular, forideals (alias filters) in permutable and weakly regular varieties see also [3] for details.
Definition 4.5. An ideal term for sectionally pseudocomplemented lattices is a term t ( x , . . . , x n , y , . . . , y m ) in the language of sectionally pseudocomplemented lattices satis-fying the identity t ( x , . . . , x n , , . . . , ≈ . Of course, there exists an infinite number of ideal terms in sectionally pseudocomple-mented lattices. The following list including five ideal terms is a so-called basis for filters in sectionally pseudocomplemented lattices, i.e. filters can be characterized by this shortlist of ideal terms.
Lemma 4.6.
The following terms are ideal terms for sectionally pseudocomplementedlattices: t := 1 ,t ( x , x , x , y , y ) := ((( x ∨ x ) ∧ ( y ∗ x ) ∧ y ) ∨ x ) ∗ ( x ∨ x ) ,t ( x , x , x , y , y ) := ((( x ∨ x ) ∧ ( y ∗ x ) ∧ y ) ∧ x ) ∗ ( x ∧ x ) ,t ( x , x , x , y , y ) := ((( x ∨ x ) ∧ ( y ∗ x ) ∧ y ) ∗ x ) ∗ ( x ∗ x ) ,t ( x , x , x , y , y ) := ( x ∗ x ) ∗ ( x ∗ ((( x ∨ x ) ∧ ( y ∗ x ) ∧ y )) . Proof.
Put t ( x, y, z, u ) := ( x ∨ y ) ∧ ( z ∗ y ) ∧ u. Then t ( x , x , x , y , y ) = ( t ( x , x , y , y ) ∨ x ) ∗ ( x ∨ x ) ,t ( x , x , x , y , y ) = ( t ( x , x , y , y ) ∧ x ) ∗ ( x ∧ x ) ,t ( x , x , x , y , y ) = ( t ( x , x , y , y ) ∗ x ) ∗ ( x ∗ x ) ,t ( x , x , x , y , y ) = ( x ∗ x ) ∗ ( x ∗ t ( x , x , y , y )) . and according to Lemma 2.3 t ( x, y, ,
1) = ( x ∨ y ) ∧ (1 ∗ y ) ∧ x ∨ y ) ∧ y = y t ( x , x , x , ,
1) = ( t ( x , x , , ∨ x ) ∗ ( x ∨ x ) = ( x ∨ x ) ∗ ( x ∨ x ) = 1 ,t ( x , x , x , ,
1) = ( t ( x , x , , ∧ x ) ∗ ( x ∧ x ) = ( x ∧ x ) ∗ ( x ∧ x ) = 1 ,t ( x , x , x , ,
1) = ( t ( x , x , , ∗ x ) ∗ ( x ∗ x ) = ( x ∗ x ) ∗ ( x ∗ x ) = 1 ,t ( x , x , x , ,
1) = ( x ∗ x ) ∗ ( x ∗ t ( x , x , , x ∗ x ) ∗ ( x ∗ x ) = 1 . The closedness with respect to ideal terms was also introduced by A. Ursini ([7]).
Definition 4.7. A subset A of a sectionally pseudocomplemented lattice L = ( L, ∨ , ∧ , ∗ ) is said to be closed with respect to the ideal terms t i ( x , . . . , x n , y , . . . , y m ) , i ∈ I , if forevery i ∈ I , all x , . . . , x n ∈ L and all y , . . . , y m ∈ A we have t i ( x , . . . , x n , y , . . . , y m ) ∈ A . Now we prove that the ideal terms listed in Lemma 4.6 form a basis for filters, i.e. filtersare characterized as those subsets which are closed with respect to these ideal terms.
Theorem 4.8.
Let L = ( L, ∨ , ∧ , ∗ ) be a sectionally pseudocomplemented lattice and F ⊆ L . Then F ∈ Fil L if and only if F is closed with respect to the ideal terms t , . . . , t listed in Lemma 4.6.Proof. If F ∈ Fil L then F = [1](Φ( F )) according to Theorem 2.5, and if t i ( x , . . . , x n , y , . . . , y m ) , i ∈ { , . . . , } , are the ideal terms listed in Lemma 4.6, a , . . . , a n ∈ L and b , . . . , b m ∈ F then t i ( a , . . . , a n , b , . . . , b m ) ∈ [ t i ( a , . . . , a n , , . . . , F )) = [1](Φ( F )) = F according to Lemma 4.6 and hence F is closed with respect to the ideal terms t , . . . , t .Conversely, assume F to be closed with respect to the ideal terms t , . . . , t . Then 1 = t ∈ F . Now assume a, b ∈ L and a ∗ b, b ∗ a ∈ F . For the term t ( x, y, z, u ) := ( x ∨ y ) ∧ ( z ∗ y ) ∧ u we have t ( x , x , x , y , y ) = ( t ( x , x , y , y ) ∨ x ) ∗ ( x ∨ x ) ,t ( x , x , x , y , y ) = ( t ( x , x , y , y ) ∧ x ) ∗ ( x ∧ x ) ,t ( x , x , x , y , y ) = ( t ( x , x , y , y ) ∗ x ) ∗ ( x ∗ x ) ,t ( x , x , x , y , y ) = ( x ∗ x ) ∗ ( x ∗ t ( x , x , y , y ))and according to Lemma 2.3 (iv) and (vi) we obtain t ( x, y, x ∗ y, y ∗ x ) = ( x ∨ y ) ∧ (( x ∗ y ) ∗ y ) ∧ ( y ∗ x ) == (( y ∨ x ) ∧ ( y ∗ x )) ∧ (( x ∗ y ) ∗ y ) = x ∧ (( x ∗ y ) ∗ y ) = x. a ∨ c ) ∗ ( b ∨ c ) = ( t ( a, b, a ∗ b, b ∗ a ) ∨ c ) ∗ ( b ∨ c ) = t ( a, b, c, a ∗ b, b ∗ a ) ∈ F, ( a ∧ c ) ∗ ( b ∧ c ) = ( t ( a, b, a ∗ b, b ∗ a ) ∧ c ) ∗ ( b ∧ c ) = t ( a, b, c, a ∗ b, b ∗ a ) ∈ F, ( a ∗ c ) ∗ ( b ∗ c ) = ( t ( a, b, a ∗ b, b ∗ a ) ∗ c ) ∗ ( b ∗ c ) = t ( a, b, c, a ∗ b, b ∗ a ) ∈ F, ( c ∗ a ) ∗ ( c ∗ b ) = ( c ∗ a ) ∗ ( c ∗ t ( b, a, b ∗ a, a ∗ b )) = t ( a, b, c, a ∗ b, b ∗ a ) ∈ F showing F ∈ Fil L . Remark 4.9.
Let us note that the term t from the proof of Theorem 4.8 gives rise to aMaltsev term. Namely, if t ( x, y, z, u ) := ( x ∨ y ) ∧ ( z ∗ y ) ∧ u and q ( x, y, z ) := t ( x, z, x ∗ y, y ∗ x ) . then q ( x, y, z ) = ( x ∨ z ) ∧ (( x ∗ y ) ∗ z ) ∧ ( y ∗ x ) ,q ( x, x, z ) = ( x ∨ z ) ∧ (( x ∗ x ) ∗ z ) ∧ ( x ∗ x ) = ( x ∨ z ) ∧ (1 ∗ z ) ∧ x ∨ z ) ∧ z = z,q ( x, z, z ) = ( x ∨ z ) ∧ (( x ∗ z ) ∗ z ) ∧ ( z ∗ x ) = (( z ∨ x ) ∧ ( z ∗ x )) ∧ (( x ∗ z ) ∗ z ) == x ∧ (( x ∗ z ) ∗ z ) = x. Observe that the Maltsev term q ( x, y, z ) is different from that in Proposition 2.1. In the following we write a ∧ b ∧ c instead of inf( a, b, c ).Now we introduce a certain modification of the notion an ideal term (for posets) whichneed not be defined everywhere. This will be used in the sequel. Definition 4.10. A partial ideal term for sectionally pseudocomplemented posets withgreatest element is a partially defined term T ( x , . . . , x n , y , . . . , y m ) in the language ofsectionally pseudocomplemented posets with greatest element satisfying the identity T ( x , . . . , x n , , . . . , ≈ . This language contains also a binary operator U ( x, y ) . Using of the concept of partial ideal terms, we will try to describe filters also in stronglysectionally pseudocomplemented posets. Similarly as in Lemma 4.6 we firstly get a listof four partial ideal terms which will be shown to suffice.
Lemma 4.11.
The following partial terms are partial ideal terms for strongly sectionallypseudocomplemented posets: T := 1 ,T ( x , x , x , y , y ) := (( U ( x , x ) ∧ ( y ∗ x ) ∧ y ) ∗ x ) ∗ ( x ∗ x ) ,T ( x , x , x , y , y ) := ( x ∗ x ) ∗ ( x ∗ ( U ( x , x ) ∧ ( y ∗ x ) ∧ y )) ,T ( x , x , x , x , y , y , y , y ) := (( U ( x , x ) ∧ ( y ∗ x ) ∧ y ) ∧∧ ( U ( x , x ) ∧ ( y ∗ x ) ∧ y )) ∗ ( x ∧ x ) . roof. Put T ( x, y, z, u ) := U ( x, y ) ∧ ( z ∗ y ) ∧ u. Then T ( x , x , x , y , y ) = ( T ( x , x , y , y ) ∗ x ) ∗ ( x ∗ x ) ,T ( x , x , x , y , y ) = ( x ∗ x ) ∗ ( x ∗ T ( x , x , y , y )) ,T ( x , x , x , x , y , y , y , y ) = ( T ( x , x , y , y ) ∧ T ( x , x , y , y )) ∗ ( x ∧ x )and according to Lemma 3.2 and Remark 3.3 T ( x, y, ,
1) = U ( x, y ) ∧ (1 ∗ y ) ∧ U ( x, y ) ∧ y = y. Hence T ( x , x , x , ,
1) = ( T ( x , x , , ∗ x ) ∗ ( x ∗ x ) = ( x ∗ x ) ∗ ( x ∗ x ) = 1 ,T ( x , x , x , ,
1) = ( x ∗ x ) ∗ ( x ∗ T ( x , x , , x ∗ x ) ∗ ( x ∗ x ) = 1 ,T ( x , x , x , x , , , ,
1) = ( T ( x , x , , ∧ T ( x , x , , ∗ ( x ∧ x ) == ( x ∗ x ) ∗ ( x ∗ x ) = 1 . Now we define closedness with respect to partial ideal terms.
Definition 4.12. A subset A of a strongly sectionally pseudocomplemented poset P =( P, ≤ , ∗ , is said to be closed with respect to the partial ideal terms T i ( x , . . . , x n , y , . . .. . . , y m ) , i ∈ I , if for every i ∈ I , all x , . . . , x n ∈ P and all y , . . . , y m ∈ A we have that T i ( x , . . . , x n , y , . . . , y m ) is defined and T i ( x , . . . , x n , y , . . . , y m ) ∈ A . Although our ideal terms are only partial, we can prove that every subset of a stronglysectionally pseudocomplemented poset P closed with respect to them is really a filter of P . Theorem 4.13.
Let P = ( P, ≤ , ∗ , be a strongly sectionally pseudocomplemented posetand F a subset of P that is closed with respect to the partial ideal terms T , . . . , T listedin Lemma 4.11. Then F ∈ Fil P .Proof. We have 1 = T ∈ F . Now assume a, b, c, d ∈ P and a ∗ b, b ∗ a, c ∗ d, d ∗ c ∈ F . Forthe partial term T ( x, y, z, u ) := U ( x, y ) ∧ ( z ∗ y ) ∧ u we have T ( x , x , x , y , y ) = ( T ( x , x , y , y ) ∗ x ) ∗ ( x ∗ x ) ,T ( x , x , x , y , y ) = ( x ∗ x ) ∗ ( x ∗ T ( x , x , y , y )) ,T ( x , x , x , x , y , y , y , y ) = ( T ( x , x , y , y ) ∧ T ( x , x , y , y )) ∗ ( x ∧ x )and according to Lemma 3.2 and Remark 3.3 we obtain T ( x, y, x ∗ y, y ∗ x ) = U ( x, y ) ∧ (( x ∗ y ) ∗ y ) ∧ ( y ∗ x ) == ( U ( y, x ) ∧ ( y ∗ x )) ∧ (( x ∗ y ) ∗ y ) = x ∧ (( x ∗ y ) ∗ y ) = x. a ∗ c ) ∗ ( b ∗ c ) = ( T ( a, b, a ∗ b, b ∗ a ) ∗ c ) ∗ ( b ∗ c ) = T ( a, b, c, a ∗ b, b ∗ a ) ∈ F, ( c ∗ a ) ∗ ( c ∗ b ) = ( c ∗ a ) ∗ ( c ∗ T ( b, a, b ∗ a, a ∗ b )) = T ( a, b, c, a ∗ b, b ∗ a ) ∈ F. Moreover, if a and c are comparable and b and d are comparable then we apply the partialterm T to derivemin( a, c ) ∗ min( b, d ) = ( T ( a, b, a ∗ b, b ∗ a ) ∧ T ( c, d, c ∗ d, d ∗ c )) ∗ ( b ∧ d ) == T ( a, b, c, d, a ∗ b, b ∗ a, c ∗ d, d ∗ c ) ∈ F. This shows F ∈ Fil P . Remark 4.14.
Let us consider the partial term T ( x, y, z, u ) := U ( x, y ) ∧ ( z ∗ y ) ∧ u fromthe proof of Lemma 4.11 and put Q ( x, y, z ) := T ( x, z, x ∗ y, y ∗ x ) , i.e. Q ( x, y, z ) = U ( x, z ) ∧ (( x ∗ y ) ∗ z ) ∧ ( y ∗ x ) . Of course, this is only a partial term because the infimum in Q need not exists for someelements from a strongly sectionally pseudocomplemented poset P = ( P, ≤ , ∗ , . It is ofsome interest that this partial term behaves like a Maltsev term. Namely, we can easilycompute Q ( x, x, z ) = U ( x, z ) ∧ (( x ∗ x ) ∗ z ) ∧ ( x ∗ x ) = U ( x, z ) ∧ (1 ∗ z ) ∧ U ( x, y ) ∧ z = z,Q ( x, z, z ) = U ( x, z ) ∧ (( x ∗ z ) ∗ z ) ∧ ( z ∗ x ) = ( U ( z, x ) ∧ ( z ∗ x )) ∧ (( x ∗ z ) ∗ z ) == x ∧ (( x ∗ z ) ∗ z ) = x. Moreover, these expressions Q ( x, x, z ) and Q ( x, z, z ) are defined for all x, z ∈ P . For every sectionally pseudocomplemented lattice L = ( L, ∨ , ∧ , ∗ ) and every M ⊆ L let F ( M ) denote the filter of L generated by M .The connection between filters generated by a certain subset and congruences on section-ally pseudocomplemenetd lattices is described in the following proposition. Proposition 4.15.
Let L = ( L, ∨ , ∧ , ∗ ) be a sectionally pseudocomplemented lattice, M ⊆ L and a ∈ L . Then Φ( F ( M )) = Θ( M × { } ) , [1](Θ( M × { } )) = F ( M ) . In particular, Φ( F ( a )) = Θ( a, , [1](Θ( a, F ( a ) . roof. Since M × { } ⊆ Φ( F ( M )) we haveΘ( M × { } ) ⊆ Φ( F ( M ))and hence [1](Θ( M × { } )) ⊆ [1](Φ( F ( M ))) = F ( M )according to Corollary 2.7. Because of M ⊆ [1](Θ( M × { } )) we have F ( M ) ⊆ [1](Θ( M × { } ))and hence Φ( F ( M )) ⊆ Φ([1](Θ( M × { } ))) = Θ( M × { } )according to Corollary 2.7.An analogous result holds for strongly sectionally pseudocomplemented posets. References [1] G. Birkhoff, Lattice Theory. AMS Colloq. Publ. (1979), Providence, R. I. ISBN0-8218-1025-1.[2] I. Chajda, An extension of relative pseudocomplementation to non-distributive lat-tices. Acta Sci. Math. (Szeged) (2003), 491–496.[3] I. Chajda, G. Eigenthaler and H. L¨anger, Congruence Classes in Universal Algebra.Heldermann, Lemgo 2012. ISBN 3-88538-226-1.[4] I. Chajda, H. L¨anger and J. Paseka, Sectionally pseudocomplemented posets. Order(submitted). http://arxiv.org/abs/1905.09343.[5] R. P. Dilworth, Non-commutative residuated lattices. Trans. Amer. Math. Soc. (1939), 426–444.[6] H. Lakser, The structure of pseudocomplemented distributive lattices. I. Subdirectdecomposition. Trans. Amer. Math. Soc. (1971), 335–342.[7] A. Ursini, Sulle variet`a di algebre con una buona teoria degli ideali. Boll. Un. Mat.Ital. (1972), 90–95.[8] H. Werner, A Mal’cev condition for admissible relations. Algebra Universalis3