Some properties of pseudoBCK and pseudoBCIalgebras
17. listopadu 12, 77146 Olomouc, Czech Republic
Abstract
PseudoBCIalgebras generalize both BCIalgebras and pseudoBCKalgebras, which are a noncommutative generalization of BCKalgebras. In this paper, following [41], we show that pseudoBCIalgebras are the residuation subreducts of semiintegral residuated pomonoids and characterize those pseudoBCIalgebras which are direct products of pseudoBCKalgebras and groups (regarded as pseudoBCIalgebras). We also show that the quasivariety of pseudoBCIalgebras is relatively congruence modular; in fact, we prove that this holds true for all relatively point regular quasivarieties which are relatively ideal determined, in the sense that the kernels of relative congruences can be described by means of ideal terms.
Keywords: PseudoBCKalgebra; pseudoBCIalgebra; filter; prefilter; ideal term; relative congruence modularity.
This is an accepted manuscript of an article published in Fuzzy Sets and Systems.
The final publication is available at
https://doi.org/10.1016/j.fss.2016.12.014.
1 Introduction
BCK and BCIalgebras were introduced by Iséki [27] as algebras induced by Meredith’s implicational logics BCK and BCI. As a matter of fact, BCKalgebras are the equivalent algebraic semantics for the logic BCK, but BCI is not algebraizable in the sense of [1]. Nevertheless, both BCK and BCIalgebras are closely related to residuated commutative pomonoids because, as is wellknown, BCKalgebras are the subreducts of integral residuated commutative pomonoids (see [17, 38, 39]), and BCIalgebras are the subreducts of semiintegral ones, where “semiintegral” means that the monoid identity is a maximal element of the underlying poset (see [41]).
Also some other algebras of logic, such as MValgebras, BLalgebras or hoops, are (equivalent to) certain integral residuated commutative pomonoids. Although noncommutative residuated pomonoids were known since the 1930s (for historical overview see [18]), noncommutative versions of the aforementioned algebras were introduced only about 15 years ago in a series of papers by Georgescu, Iorgulescu and coauthors in which they defined pseudoMValgebras^{3}^{3}3Rachůnek [40] independently introduced another noncommutative generalization of MValgebras, called GMValgebras. In fact, pseudoMValgebras and these GMValgebras are equivalent. In [18], the name “GMValgebra” has a different, more general meaning. [20], pseudoBLalgebras [8], pseudohoops [21], and pseudoBCKalgebras [19]. The prefix “pseudo” is meant to indicate noncommutativity, so that a commutative pseudoXalgebra is an Xalgebra. We refer the reader to [7] or [26] for an overview of noncommutative generalizations of algebras of logic.
PseudoBCKalgebras are algebras with two binary operations and a constant such that is a BCKalgebra whenever and coincide; the exact definition is given in Section 2. An important ingredient is the partial order which is given by iff iff . The underlying poset has no special properties except that is its greatest element. A particular case of pseudoBCKalgebras are Bosbach’s cone algebras [4] which may be thought of as the subreducts of negative cones of latticeordered groups. That pseudoBCKalgebras are indeed the subreducts of integral residuated pomonoids was proved in [35] and independently in [43] where pseudoBCKalgebras are called biresiduation algebras.
In this context we should mention Komori’s BCCalgebras [31] which are the subreducts of integral onesided residuated pomonoids (see [38]). It was noted in [49] that an algebra is a pseudoBCKalgebra if and only if both and are BCCalgebras satisfying the identity .
PseudoBCIalgebras were introduced by Dudek and Jun [9] as a noncommutative generalization of BCIalgebras. They are also algebras with the same partial order as pseudoBCKalgebras such that is a BCIalgebra when and coincide, but in contrast to pseudoBCKalgebras, is merely a maximal element of the underlying poset. Thus, roughly speaking, the difference between pseudoBCK and pseudoBCIalgebras is the same as the difference between BCK and BCIalgebras. In the present paper, we focus on some properties of BCIalgebras and try to establish analogous results in the setting of pseudoBCIalgebras. Basically, there are three topics we deal with: (1) embedding of pseudoBCIalgebras into residuated pomonoids; (2) congruence properties of the quasivarieties of pseudoBCK and pseudoBCIalgebras; and (3) direct products of pseudoBCKalgebras and groups which are regarded as pseudoBCIalgebras.
The paper is organized as follows. Section 2 is an introduction to residuated pomonoids, pseudoBCK and pseudoBCIalgebras. In Section 3, we prove that up to isomorphism every pseudoBCIalgebra is a subalgebra of the reduct of a semiintegral residuated pomonoid. Actually, we only sketch the construction which is almost identical to the one for BCIalgebras and semiintegral residuated commutative pomonoids presented by Raftery and van Alten [41].
Section 4 is devoted to congruence properties of pseudoBCK and pseudoBCIalgebras. They form relatively regular quasivarieties and ; the former is relatively congruence distributive, the latter only relatively congruence modular. In fact, most results are proved for any relatively ideal determined quasivariety , by which we mean that the kernels (classes) of relative congruences of algebras in may be characterized via ideal terms, similarly to ideal determined varieties (see [24, 2]). We prove that the relative congruence lattices of algebras in such a quasivariety are arguesian. We also briefly discuss prefilters of pseudoBCK and pseudoBCIalgebras; the name “filter” is reserved for the congruence kernels. Here, we can see an analogy with subgroups of groups: when a group is regarded as a pseudoBCIalgebra, then its subgroups are precisely the prefilters of this pseudoBCIalgebra. The study of prefilter lattices has proven to be useful for pseudoBCKalgebras as well as various classes of residuated pomonoids; see e.g. [35, 32, 10, 25, 42, 5].
2 Preliminaries
The basic concept is that of a residuated partially ordered monoid (pomonoid for short). A residuated pomonoid^{4}^{4}4It follows from (2.1) that is indeed a partially ordered monoid, in the sense that is preserved by multiplication on both sides. is a structure , where is a monoid, is a partial order^{5}^{5}5If the poset is a lattice, then —or more exactly, where are the associated lattice operations—is called a residuated monoid or a residuated lattice. on , and are binary operations on satisfying the residuation law, for all :
(2.1) 
It is clear that the monoid reduct is commutative if and only if for all , in which case is a residuated commutative pomonoid. A residuated pomonoid is said to be

semiintegral if the monoid identity is a maximal element of the poset ;

integral if is the greatest element of .
In the (semi) integral case, is specified by either of the “arrows” because by (2.1) one has
(2.2) 
Thus (semi) integral residuated pomonoids may be likewise defined as algebras of type . The class of such algebras is a quasivariety, but not a variety. The quasivariety of semiintegral residuated pomonoids can be axiomatized by
(2.3)  
(2.4)  
(2.5)  
(2.6)  
(2.7)  
(2.8) 
and the quasivariety of integral residuated pomonoids by (2.3)–(2.8) together with
(2.9) 
The proof is straightforward. In (2.8) and (2.9) we could equally use instead of , and (2.7) could be replaced by the identity .
PseudoBCKalgebras were introduced by Georgescu and Iorgulescu [19] as a noncommutative generalization of BCKalgebras; they are equivalent to biresiduation algebras which were defined by van Alten [43]. Later, pseudoBCIalgebras were defined by Dudek and Jun [9] along the same lines, thus they generalize both BCIalgebras and pseudoBCKalgebras. There are several possible definitions out of which we choose this (cf. [16] or [34]):
A pseudoBCIalgebra is an algebra of type satisfying the identities (2.3)–(2.6) and the quasiidentity (2.8), and a pseudoBCKalgebra is a pseudoBCIalgebra which in addition satisfies the identity (2.9).^{6}^{6}6As in the case of residuated pomonoids, in (2.8) and (2.9), could be equivalently replaced by . If is a pseudoBCIalgebra (resp. pseudoBCKalgebra) in which for all , then the algebra is a BCIalgebra (resp. BCKalgebra).^{7}^{7}7Iséki [27] originally defined BCI and BCKalgebras as algebras , where may be thought of as a kind of subtraction and is a minimal (or the least) element of the underlying poset which is defined by iff . Similarly, in [19] and [9], pseudoBCI and pseudoBCKalgebras were defined as algebras with two subtractions and minimal element .
For any pseudoBCIalgebra , the relation given by (2.2)—more precisely, by iff , which is equivalent to —is a partial order on . The poset is the underlying poset of . The difference between pseudoBCK and pseudoBCIalgebras is that in the former case, is the greatest element of the underlying poset, while in the latter case, is only a maximal element. Thus, pseudoBCKalgebras are integral and pseudoBCIalgebras semiintegral.
Of course, pseudoBCI and pseudoBCKalgebras are closely related to residuated pomonoids. If is a semiintegral (resp. integral) residuated pomonoid, then every subalgebra of the reduct is a pseudoBCIalgebra (resp. pseudoBCKalgebra). All pseudoBCKalgebras (biresiduation algebras) arise in this way, i.e., up to isomorphism, every pseudoBCKalgebra is a subreduct of an integral residuated pomonoid. In Section 3 we establish an analogue for pseudoBCIalgebras and semiintegral residuated pomonoids.
An easy but important observation is this: Given a pseudoBCIalgebra (resp. pseudoBCKalgebra) , the algebra is a pseudoBCIalgebra (resp. pseudoBCKalgebra), too. It is plain that and have the same underlying poset , but it can easily happen that the algebras and are not isomorphic. For example, the prelinearity identities
are independent in general. Note that if satisfies either of the two identities, then holds in , so is a pseudoBCKalgebra.
In the following lemma we list the basic arithmetical properties of pseudoBCIalgebras. For completeness, we also include some properties that have been implicitly mentioned above, such as (1), (2) or (4).
Lemma 2.1 (cf. [9], [29], [37]).
In any pseudoBCIalgebra , for all :

, ;

, ;

, ;

iff ;

implies and ;

;

iff ;

, ;

implies and ;

;

, ;

, .
Note that the inequalities and hold true only in pseudoBCKalgebras (because is equivalent to ).
Let be a pseudoBCIalgebra. The identities (11) of Lemma 2.1 entail that the set
which we call the integral part of , is a subuniverse of . Clearly, the subalgebra is a pseudoBCKalgebra; in fact, is the largest subalgebra of which is a pseudoBCKalgebra. By the group part of the pseudoBCIalgebra we mean the set
Again, by (10) and (11), is a subuniverse of . The name “group part” is justified by the fact that the subalgebra is term equivalent to a group. Indeed, if we put
for , then is a group in which is the inverse of ; the original operations and on are retrieved by
respectively. This was essentially proved in [12, 13] (also see [46, 45]).
If we define , then is a group which is isomorphic to and gives rise to the pseudoBCIalgebra , because and . Consequently, the map is an isomorphism between and (though and need not be isomorphic).
The following was independently proved in several papers; see [12, 13], [30] or [46, 45]. The proof is easy anyway.
Lemma 2.2.
For any pseudoBCIalgebra , is the set of the maximal elements of . For every , the element is the only such that . Therefore, iff .
Analogously, for the integral part we have:
Lemma 2.3.
Let be a pseudoBCIalgebra. For any , , and hence iff .
Proof.
Since , it follows . If , then . ∎
We adopt the notation of [41], but the sets and can be found in the literature under various names and symbols. is often called the pseudoBCKpart of and denoted by , and is called the antigrouped part or the center of , denoted by or , where “” comes from “atoms”. PseudoBCIalgebras satisfying are called psemisimple.
Another important consequence of Lemma 2.1 (10) and (11) is that the map
(2.10) 
is a homomorphism from onto , and
(2.11) 
is a homomorphism from onto . We have and , thus is always the kernel of a relative congruence of , while is not in general (see Section 4).
Given any pseudoBCKalgebra and any group (regarded as a pseudoBCIalgebra), we can always construct a pseudoBCIalgebra such that and . For example, we can take the direct product because and . In Section 5 we characterize those pseudoBCIalgebras which are isomorphic to the direct product . But there is an easier construction, see [47] (in fact, Theorem 4.3 therein presents a construction of a semiintegral residuated pomonoid from a bounded integral residuated pomonoid and a group).
Let be a pseudoBCKalgebra, be a group such that , and let be the pseudoBCIalgebra derived from the group, i.e. and . Let be equipped with the operations defined as follows:
where . Then is a pseudoBCIalgebra with and . The proof is but a direct inspection of all possible cases. Note that is not a subalgebra of the direct product .
3 Embedding into the reducts of residuated pomonoids
It is folklore that BCKalgebras are the subreducts of integral residuated commutative pomonoids (see [17, 38, 39]). Similarly, pseudoBCKalgebras (biresiduation algebras) are the subreducts of integral residuated pomonoids, which was proved independently by van Alten in [43] and by the second author in [33]. Also see [34, Theorem 1.2.1], the proof is just a “noncommutative modification” of the construction of [38]. In [41, Theorem 2], Raftery and van Alten proved that BCIalgebras are the subreducts of semiintegral residuated commutative pomonoids, and the aim of this section is to generalize their result to pseudoBCIalgebras.
In proving that a pseudoBCKalgebra is isomorphic to a subalgebra of the residuation reduct of an integral residuated pomonoid, one first constructs a certain pomonoid, say , whose identity is also its smallest element, and then takes the set of orderfilters of this intermediate pomonoid in order to obtain the integral residuated pomonoid into which embeds. In the case that is a pseudoBCIalgebra, the first step remains unchanged, but is merely a minimal element in the pomonoid , and hence the second step must be modified as in [41].
Let be a pomonoid in which is a minimal element. Let be the set of the minimal elements of and suppose that every exceeds a unique . Suppose further that is a group such that for all . Note that we do not require that be a subgroup of .
Let be the set of all (nonempty) orderfilters in the poset with the property that ^{8}^{8}8As usual, for an element in a poset , denotes the orderfilter . for some . Note that for every , because there is a unique with , and hence the map is an antitone injection from into . For any , let
Lemma 3.1.
If is a pomonoid satisfying the above conditions, then is a semiintegral residuated pomonoid.
Proof.
As in [41], we leave the details to the reader, we only make two remarks. First, if and where , then for all and , whence . Second, if where , then and so for any , and at the same time, . Since there is a unique minimal element below , it follows that , whence . This shows that , and so . Analogously, . ∎
Now, let be a pseudoBCIalgebra. As in Section 2, denotes the group part of . Let be the set of all nonempty words over the set . For any such and we write
In view of Lemma 2.1 (4) and (6), we have iff , and also for all . For any , let
and let . Note that for any , and hence for any . Also, iff for all , whence the map is an antitone injection from into . Further, let
thus .
Lemma 3.2.
For any pseudoBCIalgebra , is a pomonoid such that with are the minimal elements of . Every contains a unique , namely, . Moreover, for all and , if and , then , where is calculated in the group .
Proof.
Again, the proof is straightforward. We only prove the last two statements. If where , then , so whence . Since and are maximal elements, . Furthermore, if and where , then . ∎
Since is a group, it follows that also is a group, where and .^{9}^{9}9In general, for because it can happen that , but , or vice versa. Moreover, by the last statement of Lemma 3.2 we have , thus the pomonoid satisfies the assumptions of Lemma 3.1, and hence is a semiintegral residuated pomonoid. If is a pseudoBCKalgebra, then is integral.
Given , we have iff iff , whence the composite injection is given by
We conclude:
Theorem 3.3.
For any pseudoBCIalgebra , the map
is an embedding of into the pseudoBCIalgebra . Consequently, pseudoBCIalgebras (resp. pseudoBCKalgebras) are the subreducts of semiintegral (resp. integral) residuated pomonoids.
4 Relative congruences, filters and prefilters
First, we recall some general facts; see e.g. [2, 3]. Let be a quasivariety of algebras of a given language with a constant . A congruence of an algebra is a relative congruence (or a congruence) if the quotient algebra belongs to . The set of all relative congruences of is denoted by ; ordered by setinclusion, it forms an algebraic lattice . If the lattice of (relative) congruences of every algebra is modular or distributive, then is said to be (relatively) congruence modular or (relatively) congruence distributive, respectively.
Furthermore, is (relatively) regular if distinct (relative) congruences of algebras have distinct kernels; in other words, if are (relative) congruences of with , then . It is known that is relatively regular if and only if there exist binary terms such that satisfies
In this case, every relative congruence is determined by its kernel by
and the kernels of relative congruences of form an algebraic lattice which is isomorphic to ; the (inverse) isomorphism is given by . It is also known that a regular variety is always congruence modular, but a relatively regular quasivariety need not be relatively congruence modular.
Now, let and be respectively the quasivariety of pseudoBCKalgebras and the quasivariety of pseudoBCIalgebras. Both and are relatively regular (but not regular), as witnessed by the terms and where is either of the arrows and . Hence every relative congruence is determined by its kernel by: iff .
An internal characterization of the kernels of relative congruences of pseudoBCIalgebras, similar to that for pseudoBCKalgebras, can be found in [14] (see [25, 34] for pseudoBCKalgebras). Given a pseudoBCIalgebra , a subset is the kernel of a (unique) relative congruence if and only if

;

if , then ;

if , then ; and

for all , iff .
In [14], such subsets are called “closed compatible deductive systems”, but we will call them “filters” (see below). It is not hard to show that in (ii) we can use in place of and that (iv) is equivalent to the condition

for all , .
Of course, (iii) is redundant if is a pseudoBCKalgebra, and (iv) or (v) is redundant if is a BCIalgebra.
There are quite a few papers devoted to the subsets satisfying (i) and (ii); they are called “deductive systems” [14], “filters” [48] or “pseudoBCIfilters” [45], and “ideals” or “pseudoBCIideals” when the dual presentation of pseudoBCIalgebras is used [12, 29, 37]. The adjective “closed” is added when also the condition (iii), which is equivalent to being a subalgebra, is satisfied [14, 12].
Our terminology is hopefully simpler: Given a pseudoBCIalgebra , we say that is

a filter of if is the kernel of some , i.e., if satisfies the above conditions (i)–(iv), or equivalently, the conditions (i)–(iii) and (v);

a prefilter of if it satisfies (i)–(iii).
Here, we require that also (iii) be satisfied because when a group is regarded as a pseudoBCIalgebra, then the subgroups are precisely the prefilters.
We let and denote respectively the set of filters and the set of prefilters of ; it is obvious that with respect to setinclusion they form algebraic lattices, and . We know that under the mutually inverse maps and where is given by
Dymek [14] proved somewhat more; he proved that if is a “compatible deductive system” not necessarily “closed” (i.e., if satisfies (i), (ii) and (iv)), then and , with equality exactly if is “closed”.
It is known that the quasivariety of BCKalgebras is relatively congruence distributive, while the quasivariety of BCIalgebras is only relatively congruence modular (see [2, 3, 41]). It comes as no surprise that likewise is relatively congruence distributive; more directly, for any pseudoBCKalgebra , is a distributive lattice and is its complete sublattice (see [34, Corollary 2.1.12, 2.2.9] and [25]). However, can be relatively congruence modular at best, because it contains a subclass which is term equivalent to groups, so the filter lattice need not be distributive. Note that—since subgroups correspond to prefilters—the prefilter lattice need not be even modular.
In what follows, we prove that is indeed relatively congruence modular. We actually prove that this holds true for any relatively regular quasivariety which has a “nice description” of the kernels of relative congruences, by which we mean a description by means of the socalled “ideal terms” (see [24, 2]).
As before, let be a quasivariety whose language has a constant . An ideal term^{10}^{10}10Since the concept of an ideal term as well as that of an ideal obviously depends on the choice of , we should more accurately speak of ideal terms and ideals. (in variables ) is a term in the language of such that satisfies the identity
An ideal^{†}^{†}footnotemark: of an algebra is a nonempty subset of which is closed under all ideal terms, in the sense that for every ideal term one has for all and . Ordered by setinclusion, the ideals of form an algebraic lattice . It is easy to see that the ideal generated by consists precisely of the elements where is an ideal term and , .
is said to be ideal determined if for every , every ideal of is the kernel of a unique congruence , in which case the map is an isomorphism between the lattices and .
Analogously, we will say that is relatively ideal determined provided that every ideal of is the kernel of a unique relative congruence , for all . If has this property, then for all .
By [24], every ideal determined variety is congruence modular. Following the proof given in [24], we prove that every relatively ideal determined quasivariety is relatively congruence modular.
Recalling the conditions (i)–(iii) and (v), one can easily find a finite basis of ideal terms for pseudoBCK and pseudoBCIalgebras:
Proposition 4.1.
Let be a pseudoBCIalgebra. For any , the following are equivalent:

is a filter of ;

is an ideal of , i.e., is closed under all ideal terms;

and is closed under the following ideal terms:
Proof.
(1) (2). If , then for some , hence is closed under all ideal terms. (2) (3). Trivial. (3) (1). Suppose that and that is closed under the three ideal terms. If , then and also . Moreover, for any and we have and . Hence . ∎
Instead of and we could take the ideal term
Corollary 4.2.
The quasivarieties and are relatively ideal determined.
By [24, Corollary 1.5], congruence lattices of algebras in ideal determined varieties are arguesian, and hence modular. For relatively ideal determined quasivarieties we have the following analogue. For the reader’s convenience we give a proof which, however, is but a reformulation of the proof of [28, Lemma 2.1] (also see [23, Theorem IV.4.10]). Let us recall that a lattice is said to be arguesian if it satisfies the identity
where and for . An arguesian lattice is a fortiori modular; see [23, pp. 260–261].
Proposition 4.3.
Let be a relatively ideal determined quasivariety.

For every , is an arguesian lattice. Hence is relatively congruence modular.

If there is a binary term in the language of such that satisfies the identities and , then is relatively congruence distributive.
Proof.
Let be a fixed algebra in a relatively ideal determined quasivariety with a constant . First, we observe that for any and we have
(4.1) 
whence the join of the ideals and in the ideal lattice is . This is basically [24, Lemma 1.4] rephrased for relatively ideal determined classes (only ideal determined classes are considered in [24]).
Since is the ideal generated by , we have iff there is a ideal term in and elements , such that . For each , let if , and otherwise. Then
since for all , and
since for all , whence . The opposite direction is clear as . This completes the proof of (4.1).
(1) To prove that is an arguesian lattice, we take , for , and put
where
for .
Suppose that . By (4.1) there exist such that and for . Obviously, . Then . Moreover, since and , and since and . Hence . Recalling (4.1) we see that the lattice is arguesian.
(2) Let and . By (4.1) there exists such that and . Let . Since satisfies the identities and , we have
Hence and , and so . ∎
The statement (2) is a corollary of [3, Corollary 12.2.5] which says that a relatively regular quasivariety is relatively congruence distributive if there is a binary term such that satisfies the identities of (2). In the case of it suffices to take .
Corollary 4.4.
The relative congruence lattice of any pseudoBCIalgebra (resp. pseudoBCKalgebra) is arguesian (resp. distributive). Thus (resp. ) is relatively congruence modular (resp. distributive).
For an alternative proof of relative congruence distributivity of pseudoBCKalgebras, see [34, Corollary 2.1.12 and 2.2.9] or [25].
By [24, Corollary 1.9], a regular variety is ideal determined if and only if there exists a binary term such that satisfies the identities
cf. Proposition 4.3 (2). Such a term is called a subtractive term for . Unfortunately, a relatively regular quasivariety having a subtractive term may not be relatively ideal determined. For example (cf. [2, Example 7.7] or [3, Example 4.7.5]), if is the quasivariety of all torsionfree abelian groups, then the additive group of integers belongs to and the only congruences of are and . But the ideals of are precisely the subgroups of , hence is not relatively ideal determined, though it is relatively regular and satisfies the identities and , i.e., is a subtractive term for .
It was proved in [3, Theorem 12.2.6] that a relatively regular quasivariety is relatively congruence modular whenever it is conservative, in the sense that and the variety generated by satisfy the same quasiidentities of the form