Filters and congruences in sectionally pseudocomplemented lattices and posets

Together with J. Paseka we introduced so-called sectionally pseudocomplemented lattices and posets and illuminated their role in algebraic constructions. We believe that—similar to relatively pseudocomplemented lattices—these structures can serve as an algebraic semantics of certain intuitionistic logics. The aim of the present paper is to define congruences and filters in these structures, derive mutual relationships between them and describe basic properties of congruences in strongly sectionally pseudocomplemented posets. For the description of filters in both sectionally pseudocomplemented lattices and posets, we use the tools introduced by A. Ursini, i.e., ideal terms and the closedness with respect to them. It seems to be of some interest that a similar machinery can be applied also for strongly sectionally pseudocomplemented posets in spite of the fact that the corresponding ideal terms are not everywhere defined.


Introduction
The concept of a relative pseudocomplemented lattice was introduced by R. P.Dilworth (Dilworth (1939)). It was used in several branches of mathematics, e.g., as an algebraic axiomatization of intuitionistic logic (by Heyting and Brouwer) where the relative pseudocomplement is interpreted as the logical connective implication.
However, every relative pseudocomplemented lattice is distributive, see, e.g., Birkhoff (1979) and Lakser (1971). Because not every non-classical propositional calculus is necessarily distributive (for instance, the logic of quantum mechanics), it was a question whether the concept of relative pseudocomplementation can be extended in a rea-sonable way to non-distributive lattices. This was realized by the first author in Chajda (2003) by introducing sectional pseudocomplementation. Later on, the concept of sectional pseudocomplementation was extended also to posets, see Chajda et al. (2021).
In the present paper we focus on congruences and filters on sectionally pseudocomplemented lattices and posets. For lattices we can use the machinery of universal algebra (see, e.g., Chajda et al. (2012)) because sectionally pseudocomplemented lattices form a variety which is congruence permutable, congruence distributive and weakly regular. The situation with sectionally pseudocomplemented posets is a bit more complicated due to the fact that such a poset in general cannot be extended to a sectionally pseudocomplemented lattice by means of the Dedekind-MacNeille completion, see Chajda et al. (2021) for the result.
Sectionally pseudocomplemented lattices having 0 and their ideals will be the topic of one of our next studies. (Here, [b) denotes the set {x ∈ L | b ≤ x}.) In this case c is called the sectional pseudocomplement of a with respect to b and it will be denoted by a * b. We consider sectionally pseudocomplemented lattices as algebras (L, ∨, ∧, * ) of type (2, 2, 2). Every non-empty sectionally pseudocomplemented lattice has a greatest element 1, namely the algebraic constant x * x. In the following we consider only non-empty lattices.
An example of a sectionally pseudocomplemented lattice that is not relatively pseudocomplemented is N 5 depicted in Fig. 1: This lattice is not distributive and hence not relatively pseudocomplemented (see Birkhoff (1979)). The operation table for the sectional pseudocomplementation is as follows: * 0 a b c 1 0 1 1 1 1 1 a b 1 b 1 1 b c a 1 c 1 c b a b 1 1 1 0 a b c 1 This variety is congruence permutable, congruence distributive and weakly regular. A Maltsev term for congruence permutability is given by For the concept of congruence permutability, we refer the reader to Chajda et al. (2012).
Weak regularity means that every congruence on a sectionally pseudocomplemented lattice with greatest element 1 is determined by its kernel, i.e., by the congruence class [1] . Hence, our first task is to describe these classes. For this purpose we introduce the following concept: Let Fil L denote the set of all filters of L. For any subset M of L define a binary relation (M) on L as follows: The following results were proved in Chajda (2003) and Chajda et al. (2021).
The relationship between congruences and filters in sectionally pseudocomplemented lattices is illuminated in the next two theorems.
Theorem 2.4 witnesses that sectionally pseudocomplemented lattices are weakly regular.
We can prove also the converse.
Proof Let a, b, c ∈ L. Evidently, (F) is symmetric and since 1 ∈ F and x * x ≈ 1 by (i) of Lemma 2.3, it is also reflexive. Assume a * b, b * a ∈ F. Then, by Definition 2.2 Hence, (F) has the substitution property with respect to all basic operations of L. Since the variety of sectionally pseudocomplemented lattices is congruence permutable, (F) is also transitive, see, e.g., Werner's Theorem (Werner (1973)) or Corollary 3.1.13 in Chajda et al. (2012), and hence, (F) ∈ Con L. Finally, the following are equivalent: It is elementary to check that for every sectionally pseudocomplemented lattice L, (Fil L, ⊆) is a complete lattice.

Example 2.6
The sectionally pseudocomplemented lattice from Fig. 1 has the following filters: The following corollary follows from Theorems 2.4 and 2.5.
Let (L, ∨, ∧, * ) be a sectionally pseudocomplemented lattice. A deductive system of L is a 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.

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 pseudocomplemented if for all a, b ∈ P there exists a greatest element c of P satisfying This element c is called the sectional pseudocomplement a * b of a with respect to b. We write sectionally pseudocomplemented posets in the form (P, ≤, * ). A strongly sectionally pseudocomplemented poset is an ordered quadruple (P, ≤ , * , 1) such that (P, ≤, * ) is a sectionally pseudocomplemented poset with greatest element 1 satisfying the identity The following results were proved in Chajda et al. (2021).
Lemma 3.2 If P = (P, ≤, * ) is a sectionally pseudocomplemented poset with greatest element 1 and a, b, c ∈ P, then It is easy to see that every sectionally pseudocomplemented lattice is a strongly sectionally pseudocomplemented poset, and a lattice is sectionally pseudocomplemented if and only if it is sectionally pseudocomplemented as a poset.

Remark 3.3
If (P, ≤, * ) is a sectionally pseudocomplemented poset and a, b ∈ P, then which shows that there exists the infimum U (a, b) ∧ (a * b), and hence, the previous is equivalent to An example of a strongly sectionally pseudocomplemented poset which is not a lattice is visualized in Fig. 2. The operation table of * is as follows: * 0 a b c d e 1 0 1 1 1 1 1 1 1 This poset is not relatively pseudocomplemented since the relative pseudocomplement of c with respect to a does not exist.
It should be noted that there are sectionally pseudocomplemented posets which are not strongly sectionally pseudocomplemented, see, e.g., Chajda et al. (2021), but these are rather curious.
Since a sectionally pseudocomplemented poset P has only one operation, namely * , a congruence on P should satisfy the substitution property with respect to * . However, this condition is rather weak and we cannot expect to obtain a natural relationship between congruences and congruence kernels similar to that obtained for sectionally pseudocomplemented lattices in the previous section. Namely, our concept of a congruence on a strongly sectionally pseudocomplemented poset should respect also some aspects of the partial 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-
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 congruence 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.
Note that the congruences on a sectionally pseudocomplemented lattice L may not coincide with the congruences on L if it is considered only as a sectionally pseudocomplemented poset.
In analogy to the lattice case, we define Definition 3.6 Let P = (P, ≤, * , 1) be a sectionally pseudocomplemented poset with greatest element 1. A filter of P is a subset F of P containing 1 and satisfying the following conditions for all x, y, z, v ∈ P: x and z are comparable and y and v are comparable, then 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 strongly sectionally pseudocomplemented poset P, (Con P, ⊆) and (Fil P, ⊆) are complete lattices. For any subset M of P put The relationship between congruences and filters in strongly sectionally pseudocomplemented posets is illuminated in the next two theorems.
Theorem 3.7 Let P = (P, ≤, * , 1) be a strongly sectionally pseudocomplemented poset and ∈ Con L. Then, [1] ∈ Fil L and for any x, y ∈ P, and hence, using again Lemma 3.2, Since P is strongly sectionally pseudocomplemented, we have a ≤ (a * b) * b and (b * a) * a ≥ b; thus, by minstability of we conclude This shows ([1] ) = . Due to the substitution property of with respect to * and the min-stability of , we obtain [1] ∈ Fil L.
We have shown that every congruence on a strongly sectionally pseudocomplemented poset is fully determined by its 1-class [1] . Hence, we conclude Corollary 3.8 Strongly sectionally pseudocomplemented posets are weakly regular.
We can prove also the converse.

Example 3.10
The lattice of filters of the strongly sectionally pseudocomplemented poset from Fig. 2 consists of the following six filters: The corresponding Hasse diagram is depicted in Fig. 3: The following corollary follows from Theorems 3.7 and 3.9.

Properties of filters
Using the min-stability property of congruences in strongly sectionally pseudocomplemented posets, we can prove Theorem 4.1 Let P = (P, ≤, * , 1) be a strongly sectionally pseudocomplemented poset and ∈ Con P. Then, every class of is a convex subset of (P, ≤).
and hence by min-stability of we obtain We now investigate quotients P/ of strongly sectionally pseudocomplemented posets P with respect to its congruences.
Let P = (P, ≤, * , 1) be a strongly sectionally pseudocomplemented poset and ∈ Con P. We define a binary relation Recall that a 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/ , ≤ ) where P = (P, ≤, * , 1) denotes the strongly sectionally pseudocomplemented poset from Fig. 2 and the congruence on P corresponding to the filter F({d, e}) of P is isomorphic to the lattice from Fig. 1.
The following theorem was partly proved for congruences on the algebraic reduct (P, * ) in Chajda et al. (2021).
The converse inclusion follows from (i).
From (iv) we conclude that if (P, ≤) satisfies the ascending chain condition (in particular, if P is finite), then every class of has a greatest element.
The following concept is inspired by the derivation rule Modus Ponens in the non-classical logic based on a sectionally pseudocomplemented poset where * models the logical connective 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 sectionally pseudocomplemented lattices. Theorem 4.3 Let P = (P, ≤, * , 1) 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, Proof We use the fact that the filter F is the 1-class of the congruence (F).
Theorem 4.3 shows that every filter is a deductive system. However, our concept of a filter is rather complicated and it seems that not all the properties of a filter are necessary to prove this assertion. We can prove Proposition 4.4 Let P = (P, ≤, * , 1) be a strongly sectionally pseudocomplemented poset and M a subset of P containing 1 and satisfying (M * (M * x)) * x ⊆ M for all x ∈ P. Then, M is a deductive system of P.
Observe that the condition mentioned in Proposition 4.4 is just the second one of (iv) of Theorem 4.3.
For the concept of an ideal of a universal algebra which corresponds to our concept of a filter and for the concept of ideal terms, the reader is referred to Ursini (1972). In particular, for ideals (alias filters) in permutable and weakly regular varieties see also Chajda et al. (2012) for details. Definition 4.5 An ideal term for sectionally pseudocomplemented lattices is a term t(x 1 , . . . , x n , y 1 , . . . , y m ) in the language of sectionally pseudocomplemented lattices satisfying the identity t(x 1 , . . . , x n , 1, . . . , 1) ≈ 1.
Of course, there exist an infinite number of ideal terms in sectionally pseudocomplemented 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 short list of ideal terms.

Lemma 4.6
The following terms are ideal terms for sectionally pseudocomplemented lattices: t(x, y, z, u) Then, x 1 , y 2 , y 1 )). and according to Lemma 2.3 The closedness with respect to ideal terms was also introduced by A. Ursini ( Ursini (1972)).

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 1 , . . . , x n , y 1 , . . . , y m ), i ∈ I , if for every i ∈ I , all x 1 , . . . , x n ∈ L and all y 1 , . . . , y m ∈ A we have t i (x 1 , . . . , x n , y 1 , . . . , y m ) ∈ A. Now we prove that the ideal terms listed in Lemma 4.6 form a basis for filters, i.e., filters are characterized as those subsets which are closed with respect to these ideal terms. are the ideal terms listed in Lemma 4.6, a 1 , . . . , a n ∈ L and b 1 , . . . , b m ∈ F, then t i (a 1 , . . ., a n , b 1 , . . ., b m )∈[t i (a 1 , . . ., a n , 1, . . ., 1)]( (F)) = [1]( (F)) = F according to Lemma 4.6 and hence F is closed with respect to the ideal terms t 1 , . . . , t 5 . Conversely, assume F to be closed with respect to the ideal terms t 1 , . . . , t 5 . Then, 1 = t 1 ∈ F. Now assume a, b ∈ L and a * b, b * a ∈ F. For the term we have and according to Lemma 2.3 (iv) and (vi) we obtain Hence, showing F ∈ Fil L.

Remark 4.9
Let us note that the term t from the proof of Theorem 4.8 gives rise to a Maltsev term. Namely, if t(x, y, z, u) 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) which need not be defined everywhere. This will be used in the sequel.
This language contains also a binary operator U (x, y).
Using the concept of partial ideal terms, we will try to describe filters also in strongly sectionally pseudocomplemented posets. Similarly as in Lemma 4.6, we firstly get a list of four partial ideal terms which will be shown to suffice.

Lemma 4.11
The following partial terms are partial ideal terms for strongly sectionally pseudocomplemented posets:

Proof Put
T (x, y, z, u) Then, and according to Lemma 3.2 Hence, Now we define closedness with respect to partial ideal terms.
Although our ideal terms are only partial, we can prove that every subset of a strongly sectionally pseudocomplemented poset P closed with respect to them is really a filter of P.
Theorem 4.13 Let P = (P, ≤, * , 1) be a strongly sectionally pseudocomplemented poset and F a subset of P that is closed with respect to the partial ideal terms T 1 , . . . , T 4 listed in Lemma 4.11. Then, F ∈ Fil P.
Proof We have 1 = T 1 ∈ F. Now assume a, b, c, d ∈ P and a * b, b * a, c * d, d * c ∈ F. For the partial term T (x, y, z, u) and according to Lemma 3.2 and Remark 3.3 we obtain Hence, Moreover, if a and c are comparable and b and d are comparable, then we apply the partial term T 4 to derive min(a, c) * min(b, d) 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 from the proof of Lemma 4.11 and put i.e., Of course, this is only a partial term because the infimum in Q need not exists for some elements from a strongly sectionally pseudocomplemented poset P = (P, ≤, * , 1). It is of some interest that this partial term behaves like a Maltsev term. Namely, we can easily compute 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 sectionally pseudocomplemented lattices is described in the following proposition. An analogous result holds for strongly sectionally pseudocomplemented posets.
Acknowledgements Support of the research of the authors by the Austrian Science Fund (FWF), project I 4579-N, and the Czech Science Foundation (GAČR), project 20-09869L, entitled "The many facets of orthomodularity", as well as by ÖAD, project CZ 02/2019, entitled "Function algebras and ordered structures related to logic and data fusion", and, concerning the first author, by IGA, project PřF 2021 030, is gratefully acknowledged.
Author contributions Both authors contributed equally to this manuscript.
Funding Open access funding provided by Austrian Science Fund (FWF). This study was funded by the Austrian Science Fund (FWF), project I 4579-N, and the Czech Science Foundation (GAČR), project 20-09869L, as well as by ÖAD, project CZ 02/2019, and, concerning the first author, by IGA, project PřF 2021 030.

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