PSEUDO-BCK-ALGEBRAS
AND RELATED STRUCTURES
Jan K ¨uhr
Univerzita Palack´eho v Olomouci 2007
Preface
Residuation is one of the most important concepts of the theory of ordered algebraic structures which naturally arises in many other fields of mathematics. The study of abstract residuated structures has originated from the investigation of ideal lattices of commutative rings with 1. In general, a partially ordered monoid is residuated if for all a, b in its universe there exist a → b = max{c : ca ≤ b} and a b = max{c : ac ≤ b}; in other words, if for every a the translations x 7→ xa and x 7→ ax are residuated map-pings. If the multiplicative identity is the greatest element in the underlying order then the monoid is integral. Residuated structures include lattice-ordered groups and their negative cones as well as algebraic models of various propositional logics. In the logical context, the monoid operation · can be interpreted as conjunction and the residuals → and as two implications (they coincide iff the conjunction is commutative).
The notion of residuation in integral residuated partially ordered commutative monoids can be characterized by Is´eki’s BCK-algebras. Pseudo-BCK-algebras recently introduced by Georgescu and Iorgulescu are a non-commutative extension of BCK-algebras; roughly speaking, a pseudo-BCK-algebra is an algebra with binary opera-tions → and that satisfies certain axioms ensuring the resulting structure to be a BCK-algebra provided → and coincide.
It is easily verified that the residuation subreducts of integral residuated po-monoids are pseudo-BCK-algebras, and hence pseudo-BCK-algebras encompass all known alge-bras of many-valued logic together with their non-commutative extensions which can be defined in the setting of integral residuated structures. In a sense, pseudo-BCK-algebras also generalize lattice-ordered groups, because every lattice-odered group is determined by its negative cone which can be regarded as a pseudo-BCK-algebra.
The text is devided into four chapters. Of course, we begin with basic definitions and several examples which can illustrate the topic. Since we are concerned with pseudo-BCK-algebras, we skip examples of BCK-algebras that can be found in litera-ture. With the exception of pseudo-MV-algebras, we neither define non-commutative algebras related to logic which are included among residuated po-monoids. The main goal of Chapter 1 is the proof that pseudo-BCK-algebras are exactly the {→, , 1}-subreducts of integral residuated lattices, thus residuation in the non-commutative case is characterized by pseudo-algebras. We also present a generalization of BCK-logic corresponding to pseudo-BCK-algebras.
In Chapter 2, we deal with deductive systems of pseudo-BCK-algebras. We describe the kernels of relative congruences as so-called compatible deductive systems which form a complete sublattice of the lattice of all deductive systems. This allows us to characterize representable pseudo-BCK-algebras (= subdirect products of linearly
Preface iii ordered pseudo-BCK-algebras), which is one of our major results.
The third chapter is devoted mainly to pseudo-BCK-join-semilattices. We first study deductive systems in depth and show that many properties of lattices of convex `-subgroups of lattice-ordered groups can be obtained for lattices of deductive systems of pseudo-BCK-semilattices. The notion of a state plays a central role throughout the second part of Chapter 3. The definition is adopted from pseudo-BL-algebras and states retain most of expected properties, but it may happen that a linearly ordered pseudo-BCK-algebra has no states, and in case of good pseudo-BCK-semilattices, Rieˇcan states do not agree with Bosbach states.
In Chapter 4, we deal with commutative pseudo-BCK-algebras which are defined as a natural generalization of well-known commutative BCK-algebras. We focus especially on commutative pseudo-BCK-algebras with the relative cancellation property LBCK-algebras). We show that every pseudo- LBCK-algebra can be embedded into a pseudo-MV-algebra. In the final section, we prove a theorem of Cantor-Bernstein type for orthogonally σ-complete commutative pseudo-BCK-algebras.
I would like to take this opportunity to acknowledge cooperation with Ivan Chajda, Radom´ır Halaˇs and Jiˇr´ı Rach˚unek. I am also grateful to my parents for their constant encouragement and patience.
Contents
1 Elements of pseudo-BCK-algebras 1
1.1 Definitions and examples . . . 1
1.2 Embedding into residuated lattices . . . 7
1.3 Pseudo-BCK-logic . . . 12
2 Deductive systems and congruences 16 2.1 The lattice of deductive systems . . . 16
2.2 Compatible deductive systems . . . 24
2.3 Representable pseudo-BCK-algebras . . . 29
3 Pseudo-BCK-semilattices 36 3.1 Congruence kernels . . . 36
3.2 More on the lattice DS(A) . . . . 39
3.3 Spectral topology . . . 44
3.4 States on bounded pseudo-BCK-semilattices . . . 47
3.5 State space . . . 55
3.6 Good pseudo-BCK-semilattices . . . 60
4 Commutative pseudo-BCK-algebras 65 4.1 Basic properties . . . 65
4.2 Relative cancellation property . . . 69
4.3 Some properties of pseudo- LBCK-algebras . . . 73
4.4 Cantor-Bernstein theorem . . . 79
Bibliography 86
List of symbols 89
Index 91
Chapter 1
Elements of
pseudo-BCK-algebras
In this introductory chapter, we first give basic definitions and examples of pseudo-BCK-algebras with emphasis on pseudo-MV-algebras, and show that the class of pseudo-BCK-algebras is a proper quasivariety. Special attention is then devoted to the relation with integral residuated lattices and it is proved that all pseudo-BCK-algebras are subreducts of (complete) integral residuated lattices. We also present a logical system which is a non-commutative version of BCK-logic and whose equivalent algebraic semantics is the quasivariety of pseudo-BCK-algebras.
1.1
Definitions and examples
BCK-algebras introduced by Is´eki [38], [39] as algebras with a binary operation ∗ (cap-turing basic features of set-theoretical difference) and constant 0 which is the least element in the induced partial order are frequently considered as algebras with a dual operation → and constant 1 which is the greatest element in the induced order of the algebra, because the dual approach makes apparent the connections with propositional logic and commutative residuated po-monoids. The same situation arises in case of pseudo-BCK-algebras. The original definition can be phrased as follows:
Definition 1.1.1[26] A pseudo-BCK-algebra is a structure (A, ≤, , ;, 0) where (A, ≤) is a poset with a least element 0, and , ; are binary operations on A such that, for all x, y, z ∈ A, we have
(z y) ; (z x) ≤ x y, (z ; y) (z ; x) ≤ x ; y, x (x ; y) ≤ y, x ; (x y) ≤ y,
x ≤ y iff x y = 0 iff x ; y = 0.
Since the partial order ≤ is determined by either of the two difference operations , ;, we can eliminate ≤ from the signature and treat pseudo-BCK-algebras as alge-bras (A, , ;, 0) of type h2, 2, 0i. A bounded pseudo-BCK-algebra is one which pos-sesses a greatest element; we shall consider bounded pseudo-BCK-algebras as algebras (A, , ;, 0, 1), 1 denoting the top element of (A, ≤).
It is clear that every BCK-algebra is a pseudo-BCK-algebra in which = ;. Hence an arbitrary poset (P, ≤) with a least element 0 can be made into a (pseudo-)BCK-algebra by putting a b = a ; b = 0 if a ≤ b, and a b = a ; b = a otherwise.
Let us present here several examples of proper pseudo-BCK-algebras:
Example 1.1.2 Any ordinal number α is a pseudo-BCK-algebra with respect to its natural order and the operations , ; given by x y = min{z ∈ α : z + y ≥ x} and x ; y = min{z ∈ α : y + z ≥ x}. We have = ; iff α ≤ ω + 1.
Example 1.1.3 Given any lattice-ordered group (G, +, −, 0, ∨, ∧), we may regard its positive cone G+ = {g ∈ G : 0 ≤ g} equipped with the order ≤ associated with the lattice operations ∨, ∧ as a pseudo-BCK-algebra where
g h = g − (g ∧ h) = (g − h) ∨ 0 and g ; h = −(g ∧ h) + g = (−h + g) ∨ 0. Obviously, = ; if the group addition + is commutative. For background on `-groups we refer to [1] or [29].
Example 1.1.4A pseudo-MV-algebra (or a GMV-algebra) (A, ⊕,−,∼, 0, 1) is a monoid
(A, ⊕, 0) endowed with a constant 1 and two supplementary unary operations satisfying the equations: x ⊕ 1 = 1 = 1 ⊕ x, 1−= 0 = 1∼, (x−⊕ y−)∼= (x∼⊕ y∼)−, x ⊕ (y · x∼) = y ⊕ (x · y∼) = (y−· x) ⊕ y = (x−· y) ⊕ x, (x−⊕ y) · x = y · (x ⊕ y∼), x−∼= x,
where the binary operation · is defined by x · y = (x−⊕ y−)∼. Pseudo-MV-algebras generalize MV-algebras in such a way that if ⊕ is commutative then the two nega-tions coincide and (A, ⊕,−, 0, 1) is an MV-algebra (see [27] and [61]). Every
pseudo-MV-algebra bears a natural lattice order that is given by x ≤ y iff x− ⊕ y = 1 (or, equivalently, if y ⊕ x∼= 1); the induced lattice is distributive and we have
x ∨ y = y ⊕ (x · y∼) = (y−· x) ⊕ y and x ∧ y = x · (y ⊕ x∼) = (x−⊕ y) · x. As proved in [26], pseudo-MV-algebras are equivalent to bounded pseudo-BCK-algebras satisfying the identities
x ; (x y) = y ; (y x),
x (x ; y) = y (y ; x). (1.1.1)
Indeed, for any pseudo-MV-algebra (A, ⊕,−,∼, 0, 1), if we define the operations , ; by
1.1. Definitions and examples 3 then (A, , ;, 0, 1) is a bounded pseudo-BCK-algebra. Conversely, if (A, , ;, 0, 1) is a bounded pseudo-BCK-algebra satisfying (1.1.1), then upon defining
x ⊕ y = 1 [(1 ; x) ; y] = 1 ; [(1 y) x], x−= 1 x and x∼ = 1 ; x,
the structure (A, ⊕,−,∼, 0, 1) is a pseudo-MV-algebra. The correspondence is
one-to-one.
Though there exist natural examples of pseudo-BCK-algebras, in many other rele-vant partially ordered structures we should have to reverse the initial order so as to obtain a pseudo-BCK-algebra in the sense of Definition 1.1.1 (for instance, in the al-gebraic model of a given propositional logic the partial order should naturally reflect deducibility in the logic). Accordingly, we shall mostly use the dual formulation which is due to A. Iorgulescu:
Definition 1.1.5 [36] A pseudo-BCK-algebra1 is a structure (A, ≤, →, , 1) where (A, ≤) is a poset with a greatest element 1, and →, are binary operations on A such that, for all x, y, z ∈ A, we have
x → y ≤ (y → z) (x → z), x y ≤ (y z) → (x z), (1.1.2)
x ≤ (x → y) y, x ≤ (x y) → y, (1.1.3)
x ≤ y iff x → y = 1 iff x y = 1. (1.1.4)
The underlying order ≤ can be retrieved via (1.1.4) and hence we may equivalently regard (A, ≤, →, , 1) to be an algebra (A, →, , 1). We shall prove later that pseudo-BCK-algebras as algebras (A, →, , 1) of type h2, 2, 0i form a quasivariety which is not a variety.
By a bounded pseudo-BCK-algebra we mean an algebra (A, →, , 0, 1) such that (A, →, , 1) is a pseudo-BCK-algebra with least element 0.
A BCK-algebra is a pseudo-BCK-algebra in which the operations → and coincide. In particular, every poset (P, ≤) with a greatest element 1 is a (pseudo-)BCK-algebra where
x → y = x y = (
1 if x ≤ y, y otherwise.
The BCK-algebra so obtained will be referred to as the BCK-algebra associated to the poset (P, ≤).
Many important examples come from what is called “algebras of logic”. Typically the operations → and can be interpreted as truth functions of the implication con-nectives of a logical system and 1 as “the true”. Algebras that arise from logic usually are definitionally equivalent to certain bounded residuated lattices (Boolean algebras, Heyting algebras, MV- and pseudo-MV-algebras, BL- and pseudo-BL-algebras, etc.) or can be isomorphically embedded into the implication reducts of certain residuated structures (Tarski algebras, Hilbert algebras, etc.). The following example is therefore fundamental:
Example 1.1.6 An integral residuated lattice [46] is an algebra (L, ∨, ∧, ·, →, , 1) such that (L, ∨, ∧) is a lattice, (L, ·, 1) is a monoid whose identity 1 is the greatest element of the lattice, and
a · b ≤ c iff a ≤ b → c iff b ≤ a c (1.1.5)
for all a, b, c ∈ L. The residuals → and coincide iff the multiplication · is commuta-tive. A straightforward verification yields that (L, →, , 1) is a pseudo-BCK-algebra.
As we shall see in Theorem 1.2.1, every pseudo-BCK-algebra is isomorphic to a subalgebra of (L, →, , 1) for a suitable integral residuated lattice.
More generally:
Example 1.1.7 A porim (= partially ordered residuated integral monoid) is a structure (P, ≤, ·, →, , 1) where (P, ≤) is a poset, (P, ·, 1) is a monoid whose identity 1 is the greatest element of (P, ≤), and the condition (1.1.5) holds for all a, b, c ∈ P . Again, → and coincide iff · is commutative. For any porim, (P, →, , 1) is a pseudo-BCK-algebra. Porims can be described as pseudo-BCK-algebras with the condition (P), i.e., pseudo-BCK-algebras expanded by a binary operation · which satisfies the equation
(x · y) → z = x → (y → z).
It is worth noticing that there exist pseudo-BCK-algebras which do not admit such a multiplication (e.g. Example 1.2.6).
Example 1.1.8 Pseudo-MV-algebras are especially nice, because with respect to the natural order given by x ≤ y iff x−⊕y = 1 they can be viewed as pseudo-BCK-algebras
in the sense both of Definition 1.1.1 (see Example 1.1.4) and of Definition 1.1.5. Indeed, for any a pseudo-MV-algebra (A, ⊕,−,∼, 0, 1), the rules
x → y = x−⊕ y and x y = y ⊕ x∼
define a bounded pseudo-BCK-algebra (A, →, , 0, 1) which, in addition, satisfies the identities
(x → y) y = (y → x) x,
(x y) → y = (y x) → x. (1.1.6)
The reverse passage from a bounded pseudo-BCK-algebra (A, →, , 0, 1) satisfying (1.1.6) to (A, ⊕,−,∼, 0, 1) is given by
x ⊕ y = (x 0) → y = (y → 0) x, x−= x → 0 and x∼= x 0. The correspondence is one-to-one. Unbounded pseudo-BCK-algebras satisfying the identities (1.1.6) are studied in depth in Chapter 4 under the name commutative pseudo-BCK-algebras.
Example 1.1.9 (cf. Example 1.1.3) Let (G, ·,−1, 1, ∨, ∧) be an `-group with the
nega-tive cone G− = {g ∈ G : g ≤ 1}. Then (G−, →, , 1), where
g → h = h · (g ∨ h)−1 and g h = (g ∨ h)−1· g, is a pseudo-BCK-algebra.
1.1. Definitions and examples 5 In addition to (1.1.2)—(1.1.4), pseudo-BCK-algebras satisfy the following easily derivable properties (see [26] or [36]) that will be used without explicit references:
x → 1 = x 1 = 1, 1 → x = 1 x = x, x → x = x x = 1, x ≤ y ⇒ z → x ≤ z → y & z x ≤ z y, x ≤ y ⇒ y → z ≤ x → z & y z ≤ x z, x (y → z) = y → (x z), x ≤ y → z ⇔ y ≤ x z, x ≤ y → x, x ≤ y x, ((x → y) y) → y = x → y, ((x y) → y) y = x y, x → y ≤ (z → x) → (z → y), x y ≤ (z x) (z y).
Moreover, if the supremumWi∈Ixiexists in the underlying poset, then so do the infima
V
i∈I(xi→ y) and
V
i∈I(xi y) and we have
_ i∈I xi → y =^ i∈I (xi → y) and _ i∈I xi y =^ i∈I (xi y).
As we have anticipated, the class of pseudo-BCK-algebras is a quasivariety: Theorem 1.1.10 An algebra (A, →, , 1) of type h2, 2, 0i is a pseudo-BCK-algebra if and only if it satisfies the following identities and quasi-identity: 2
(x → y) [(y → z) (x → z)] = 1, (1.1.7) (x y) → [(y z) → (x z)] = 1, (1.1.8) 1 → x = x, (1.1.9) 1 x = x, (1.1.10) x → 1 = 1, (1.1.11) x → y = 1 & y → x = 1 ⇒ x = y. (1.1.12)
Proof: Every pseudo-BCK-algebra certainly fulfils (1.1.7)—(1.1.12). Conversely, assume that an algebra (A, →, , 1) satisfies (1.1.7)—(1.1.12). By (1.1.7) and (1.1.9) we have x ((x → y) y) = (1 → x) ((x → y) (1 → y)) = 1, and similarly, x → ((x y) → y) = (1 x) → ((x y) → (1 y)) = 1 by (1.1.8) and (1.1.10). In particular, x x = x ((x → x) x) = 1 and x → x = x → ((x x) → x = 1. Further, if x → y = 1 then x y = x ((x → y) y) = 1, and analogously, if x y = 1 then x → y = x → ((x y) → y) = 1. Thus x → y = 1 iff x y = 1. It is therefore easily seen that the relation ≤ defined by
x ≤ y ⇔ x → y = 1
is a partial order making the structure (A, ≤, →, , 1) into a pseudo-BCK-algebra.
2The same identities and quasi-identity were used by van Alten [68] as an axiomatization of his biresiduation algebras which are, by definition, the {→, , 1}-subreducts of integral residuated lattices.
Since pseudo-BCK-algebras include BCK-algebras, which are not closed under ho-momorphic images, it follows that the quasivariety of pseudo-BCK-algebras is not a variety (for a proper BCK-algebra whose homomorphic image is not a pseudo-BCK-algebra see Example 2.2.3).
Although the underlying poset of a pseudo-BCK-algebra, in general, enjoys no particular properties (except that 1 is a greatest element), it may happen to be a semilattice or even a lattice, which is the case we are often interested in.
Definition 1.1.11 A pseudo-BCK-join-semilattice is an algebra (A, ∨, →, , 1) where (A, ∨) is a join-semilattice, (A, →, , 1) is a pseudo-BCK-algebra, and a ∨ b = b iff a → b = 1 for all a, b ∈ A.
In other words, the induced poset (A, ≤) of (A, →, , 1) is a join-semilattice with the associated join operation ∨. It can be easily seen that an algebra (A, ∨, →, , 1) of type h2, 2, 2, 0i is a pseudo-BCK-join-semilattice if and only if (A, ∨) is a join-semilattice and it satisfies the identities (1.1.7)—(1.1.11) and
x ∨ [(x → y) y] = (x → y) y, (1.1.13)
x → (x ∨ y) = 1. (1.1.14)
Therefore, the class of all pseudo-BCK-join-semilattices forms a variety.
Definition 1.1.12 A pseudo-BCK-meet-semilattice is an algebra (A, ∧, →, , 1) such that (A, ∧) is a meet-semilattice, (A, →, , 1) is a pseudo-BCK-algebra and a ∧ b = a iff a → b = 1.
As can be easily shown, an algebra (A, ∧, →, , 1) of type h2, 2, 2, 0i is a pseudo-BCK-meet-semilattice if and only if (A, ∧) is a meet-semilattice and it satisfies the identities (1.1.7)—(1.1.11) together with
x ∧ [(x → y) y] = x, (1.1.15)
(x ∧ y) → y = 1. (1.1.16)
As a particular kind of pseudo-BCK-meet-semilattices we mention pseudo-hoops (see [28]) that are naturally ordered porims, in the sense that y ≤ x iff y = x · u = v · x for some u, v. Indeed, if (A, ≤, ·, →, , 1) is a pseudo-hoop, then (A, ∧, →, , 1) is a pseudo-BCK-meet-semilattice in which x ∧ y = x · (x y) = (x → y) · x.
Definition 1.1.13 An algebra (A, ∨, ∧, →, , 1) is called a pseudo-BCK-lattice if (A, ∨, ∧) is a lattice, (A, →, , 1) is a pseudo-BCK-algebra, and a → b = 1 iff a ∨ b = b (iff a ∧ b = a) for all a, b ∈ A.
Like pseudo-BCK-semilattices, also pseudo-BCK-lattices form a variety that is ax-iomatized by the defining identities of lattices and by the identities (1.1.7)—(1.1.11), (1.1.13) and (1.1.14), or by (1.1.7)—(1.1.11), (1.1.15) and (1.1.16), respectively.
1.2. Embedding into residuated lattices 7
1.2
Embedding into residuated lattices
It is commonly known that BCK-algebras are exactly the {→, 1}-subreducts of com-mutative integral residuated lattices (see [58], [23], [60]). The purpose of the present section is to show that pseudo-BCK-algebras are the {→, , 1}-subreducts of (non-commutative) integral residuated lattices. One direction is straightforward: If we are given an arbitrary integral residuated lattice (L, ∨, ∧, ·, →, , 1) then every sub-algebra of the reduct (L, →, , 1) is a pseudo-BCK-sub-algebra and every subsub-algebra of (L, ∨, →, , 1) is a pseudo-BCK-join-semilattice. The converse was proved in [51] and slightly improved in [48]:
Theorem 1.2.1 Every pseudo-BCK-algebra can be isomorphically embedded into the {→, , 1}-reduct of a complete integral residuated lattice in such a way that also ex-isting finite suprema are preserved. Hence every pseudo-BCK-join-semilattice can be isomorphically embedded into the {∨, →, , 1}-reduct of a complete integral residuated lattice.
The proof we present here is essentially based on the embedding construction from [58]. An independent proof can be found in [68].
Given a fixed (but arbitrary) pseudo-BCK-algebra (A, →, , 1), we let W denote the set of all words
x = x1. . . xn (n ∈ N)
over A. As usual, xR and xy stand for the reverse word of x ∈ W and for the con-catenation of x, y ∈ W , respectively. For any word x = x1. . . xn ∈ W and an element
a ∈ A we shall write
x → a = x1 → (· · · → (xn→ a) · · · ) and x a = x1 (· · · (xn a) · · · ).
Observe that
x → a = 1 iff xR a = 1.
Let Q be the set of all finite non-empty subsets of W . One readily sees that the relation ∼ defined by the stipulation {x1, . . . , xm} ∼ {y1, . . . , yn} iff for all w ∈ W and a ∈ A we have
w → (xi → a) = 1 for all i = 1, . . . , m ⇔ w → (yj → a) = 1 for all j = 1, . . . , n
is an equivalence on Q; the ∼-class containing {x1, . . . , xn} will be briefly denoted as
hx1, . . . , xni.
Further, we equip the quotient set P = Q/ ∼ with two binary operations u and ∗, as follows:
hx1, . . . , xmi u hy1, . . . , yni = hx1, . . . , xm, y1, . . . , yni,
hx1, . . . , xmi ∗ hy1, . . . , yni = hxiyj : i = 1, . . . , m, j = 1, . . . , ni.
The definition of u is obviously correct. In order to show that so is the definition of the multiplication ∗ suppose hx1, . . . , xmi = hu1, . . . , uri and hy1, . . . , yni = hv1, . . . , vsi.
For arbitrary w ∈ W and a ∈ A, if
then
w → (uk → (yj → a)) = 1 for all k = 1, . . . , r, j = 1, . . . , n
by the assumption hx1, . . . , xmi = hu1, . . . , uri, and similarly
w → (uk→ (v`→ a)) = 1 for all k = 1, . . . , r, ` = 1, . . . , s
since hy1, . . . , yni = hv1, . . . , vsi. Thus
hx1, . . . , xmi ∗ hy1, . . . , yni = hu1, . . . , uri ∗ hv1, . . . , vsi.
By a meet-semilattice-ordered monoid we mean an algebra (M, ∧, ·, e) such that (M, ∧) is a meet-semilattice, (M, ·, e) is a monoid, and (x ∧ y) · z = (x · z) ∧ (y · z) and z · (x ∧ y) = (z · x) ∧ (z · y) for all x, y, z ∈ M . If the identity element e is the least element of M , then M is called dually integral.
Lemma 1.2.2 For every pseudo-BCK-algebra (A, →, , 1), the structure (P, u, ∗, h1i) is a dually integral meet-semilattice-ordered monoid.
Proof: It is evident that (P, u) is a semilattice with least element h1i since for every hx1, . . . , xni ∈ P ,
hx1, . . . , xni u h1i = hx1, . . . , xn, 1i = h1i.
Indeed, given any w ∈ W , a ∈ A, if w → (1 → a) = w → a = 1 then w → (x → a) = 1 for every x ∈ W , because w → (x → a) ≥ w → a.
The multiplication ∗ is associative and for every hx1, . . . , xni ∈ P we have hx1, . . . , xni ∗ h1i = hxi1 : i = 1, . . . , ni = hx1, . . . , xni
since w → (xi1 → a) = w → (xi → (1 → a)) = w → (xi → a) for all w ∈ W , a ∈ A.
Similarly h1i ∗ hx1, . . . , xni = hx1, . . . , xni.
There remains to be shown that ∗ distributes over u. For every hx1, . . . , xmi,
hy1, . . . , yni, hz1, . . . , zqi ∈ P we have (hx1, . . . , xmi u hy1, . . . , yni) ∗ hz1, . . . , zqi = = hx1, . . . , xm, y1, . . . , yni ∗ hz1, . . . , zqi = h{xizk: i = 1, . . . , m, k = 1, . . . , q} ∪ {yjzk : j = 1, . . . , n, k = 1, . . . , q}i = (hx1, . . . , xmi ∗ hz1, . . . , zqi) u (hy1, . . . , yni ∗ hz1, . . . , zqi), and similarly hz1, . . . , zqi ∗ (hx1, . . . , xmi u hy1, . . . , yni) = (hz1, . . . , zqi ∗ hx1, . . . , xmi) u (hz1, . . . , zqi ∗ hy1, . . . , yni).
It is worth noticing that for the partial order v associated with the meet operation u we have: hx1, . . . , xmi v hy1, . . . , yni iff for all w ∈ W and a ∈ A,
w → (xi→ a) = 1 for all i = 1, . . . , m ⇒ w → (yj → a) = 1 for all j = 1, . . . , n.
For (M, ∧, ·, e) a dually integral meet-semilattice-ordered monoid let FM be the
set of all filters of (M, ∧) augmented by ∅, i.e., FM consists of possibly empty subsets X ⊆ M such that
1.2. Embedding into residuated lattices 9 (i) x1∧ x2 ∈ X for all x1, x2∈ X,
(ii) if x ∈ X and y ≥ x then y ∈ X.
Let us introduce the following notations for X, Y ∈ FM:
X ∨ Y = {a ∈ M : a ≥ x ∧ y for some x, y ∈ X ∪ Y }, X · Y = {a ∈ M : a ≥ x · y for some x ∈ X, y ∈ Y }, X → Y = {a ∈ M : {a} · X ⊆ Y },
X Y = {a ∈ M : X · {a} ⊆ Y }.
(1.2.1)
For convenience, we shall write a·X and X ·a instead of {a}·X and X ·{a}, respectively. Lemma 1.2.3 (FM, ∨, ∩, ·, →, , ∅, M ) is a complete integral residuated lattice.
Proof: Clearly, FM partially ordered by set-theoretical inclusion is a complete
lat-tice with ∨ and ∩ (set-theoretical intersection) as the join and meet, respectively. Let X, Y ∈ FM. In order to see that X · Y satisfies the above condition (i), assume
a, b ∈ X · Y , i.e., a ≥ x1· y1 and b ≥ x2· y2 for some x1, x2 ∈ X and y1, y2 ∈ Y . Then a ∧ b ≥ (x1· y1) ∧ (x2· y2) ≥ (x1∧ x2) · (y1∧ y2), where x1∧ x2 ∈ X and y1∧ y2 ∈ Y , so a ∧ b ∈ X · Y . The condition (ii) is obvious, hence X · Y ∈ FM.
Now we show that X → Y ∈ FM. For (i), let a, b ∈ X → Y . If c ∈ (a ∧ b) · X
then c ≥ (a ∧ b) · x = (a · x) ∧ (b · x) for some x ∈ X. Since a · x ∈ a · X ⊆ Y and b · x ∈ b · X ⊆ Y , we have (a · x) ∧ (b · x) ∈ Y whence c ∈ Y . Thus (a ∧ b) · X ⊆ Y , which entails a ∧ b ∈ X · Y . The condition (ii) easily follows from the fact that a ≤ b implies b · X ⊆ a · X. The argument for X Y ∈ FM is parallel.
Let X, Y, Z ∈ FM. If a ∈ (X · Y ) · Z then a ≥ b · z for some b ∈ X · Y and z ∈ Z,
where b ≥ x · y for some x ∈ X, y ∈ Y . Hence a ≥ (x · y) · z = x · (y · z) and, since y · z ∈ Y · Z, it follows a ∈ X · (Y · Z), thus (X · Y ) · Z ⊆ X · (Y · Z). The proof of the other inclusion is analogous, so that (X · Y ) · Z = X · (Y · Z).
Further, if a ∈ X · M then a ≥ x · b for some x ∈ X and b ∈ M , and since e is the least element of M , we have a ≥ x · b ≥ x · e = x, which yields a ∈ X, proving X · M ⊆ X. From x ≥ x · e for all x ∈ X we also obtain X ⊆ X · M , so X · M = X. Similarly M · X = X.
We have proved that (FM, ·, M ) is a monoid. To complete the proof, we must verify
the residuation equivalences
X · Y ⊆ Z iff X ⊆ Y → Z iff Y ⊆ X Z.
Suppose that X ⊆ Y → Z. Let a ∈ X · Y , i.e., a ≥ x · y for some x ∈ X, y ∈ Y . It is plain that a ∈ x · Y , and by hypothesis we have x ∈ Y → Z, so x · Y ⊆ Z. It follows a ∈ Z and X · Y ⊆ Z. Conversely, assume X · Y ⊆ Z. Then for every x ∈ X, x · Y ⊆ Z, and hence x ∈ Y → Z proving X ⊆ Y → Z. One similarly proves that X · Y ⊆ Z is equivalent to Y ⊆ X Z.
Lemmata 1.2.2 and 1.2.3 allow us to construct a complete integral residuated lattice whose {→, , 1}-reduct contains an isomorphic copy of (A, →, , 1) as a subalgebra.
Let (P, u, ∗, h1i) be the meet-semilattice-ordered monoid associated to our fixed pseudo-BCK-algebra (A, →, , 1) according to Lemma 1.2.2, let (FP, ∨, ∩, ∗, →, , ∅, P ) be
the residuated lattice assigned to (P, u, ∗, h1i) by Lemma 1.2.3, and for any a ∈ A put f (a) = {hx1, . . . , xni ∈ P : xi → a = 1 for all i = 1, . . . , n}. (1.2.2)
Observe that f (a) belongs to FP, because
(i) hx1, . . . , xmi u hy1, . . . , yni ∈ f (a) whenever hx1, . . . , xmi, hy1, . . . , yni ∈ f (a),
(ii) if hx1, . . . , xmi ∈ f (a) and hx1, . . . , xmi v hy1, . . . , yni, then 1 → (xi → a) =
1 → 1 = 1 for all i = 1, . . . , m, which yields yj → a = 1 → (yj → a) = 1 for all
j = 1, . . . , n, so that hy1, . . . , yni ∈ f (a).
Proposition 1.2.4 The map f : A → FP defined by (1.2.2) is an isomorphic
em-bedding of (A, →, , 1) into (FP, →, , P ). Moreover, if a ∨ b exists in (A, ≤) then
f (a ∨ b) = f (a) ∨ f (b).
Proof: (i) If f (a) = f (b) for some a, b ∈ A, then hai ∈ f (a) and hbi ∈ f (b) imply a → b = 1 and b → a = 1, hence a = b. Thus f is injective. It is also evident that f (1) = P since x → 1 = 1 for each x ∈ W .
(ii) We show next that f preserves →, i.e., f (a → b) = f (a) → f (b) for all a, b ∈ A. Let hx1, . . . , xmi ∈ f (a → b). Then for every hy1, . . . , yni ∈ f (a),
hx1, . . . , xmi ∗ hy1, . . . , yni = hxiyj : i = 1, . . . , m, j = 1, . . . , ni ∈ f (b). (1.2.3)
Indeed, for every j = 1, . . . , n, a ≤ (a → b) b yields the inequality 1 = yj → a ≤
yj → ((a → b) b) = (a → b) (yj → b), so (a → b) (yj → b) = 1 and a → b ≤ yj → b, whence 1 = xi → (a → b) ≤ xi → (yj → b) = xiyj → b, thus
xiyj → b = 1 for all i = 1, . . . , m, j = 1, . . . , n.
It follows that hx1, . . . , xmi ∗ f (a) ⊆ f (b): if hz1, . . . , zqi ∈ hx1, . . . , xmi ∗ f (a), i.e., hz1, . . . , zqi w hx1, . . . , xmi ∗ hy1, . . . , yni for some hy1, . . . , yni ∈ f (a), then
hz1, . . . , zqi ∈ f (b). We conclude hx1, . . . , xmi ∈ f (a) → f (b), proving f (a → b) ⊆
f (a) → f (b).
Conversely, hx1, . . . , xmi ∈ f (a) → f (b) together with hai ∈ f (a) implies hxia : i =
1, . . . , mi = hx1, . . . , xmi ∗ hai ∈ f (b), so xi → (a → b) = xia → b = 1 for every i, hence
hx1, . . . , xmi ∈ f (a → b), which settles f (a) → f (b) ⊆ f (a → b).
(iii) For completeness we include the proof that f (a b) = f (a) f (b) for all a, b ∈ A, though it is similar to (ii). Let hx1, . . . , xmi ∈ f (a b). Like in the previous
claim we first prove that
hy1, . . . , yni ∗ hx1, . . . , xmi = hyjxi : j = 1, . . . , n, i = 1, . . . , mi ∈ f (b) (1.2.4)
for every hy1, . . . , yni ∈ f (a). Essential to the proof is the equivalence yj → a = 1 iff
yRj a = 1; the rest is parallel to the argument for (1.2.3). We have 1 = yRj a ≤ yR
j ((a b) → b) = (a b) → (yRj b), so (a b) → (yRj b) = 1 and
a b ≤ yRj b, whence 1 = xi → (a b) ≤ xi → (yRj b) = yRj (xi → b)
for all i = 1, . . . , m, j = 1, . . . , n. Thus yR
j (xi → b) = 1 and, equivalently,
yjxi → b = yj → (xi → b) = 1. The desired inclusion f (a) ∗ hx1, . . . , xmi ⊆ f (b) is an easy consequence of (1.2.4).
1.2. Embedding into residuated lattices 11 Conversely, assume that hx1, . . . , xmi ∈ f (a) f (b). From hai ∈ f (a) we obtain
haxi : i = 1, . . . , mi = hai ∗ hx1, . . . , xmi ∈ f (b), therefore a → (xi → b) = axi → b = 1
and xi → (a b) = a (xi → b) = 1, so hx1, . . . , xmi ∈ f (a b).
(iv) Finally, we must show that f respects finite joins. For that purpose we observe that if a ∨ b exists in A then
hai u hbi = ha, bi v hx1, . . . , xni (1.2.5)
for hx1, . . . , xni ∈ f (a ∨ b), i.e., xi → (a ∨ b) = 1 for every i = 1, . . . , n. Let w ∈ W
and c ∈ A be arbitrary. Let w → (a → c) = 1 and w → (b → c) = 1. Then a → (wR c) = wR (a → c) = 1 and b → (wR c) = wR (b → c) = 1, so a ≤ wR c and b ≤ wR c whence a ∨ b ≤ wR c. Thus (a ∨ b) → (wR c) = 1,
which is equivalent to w → ((a ∨ b) → c) = 1. Now a ∨ b ≤ ((a ∨ b) → c) c implies 1 = xi → (a ∨ b) ≤ xi → (((a ∨ b) → c) c) = ((a ∨ b) → c) (xi → c), hence
(a ∨ b) → c ≤ xi → c, which yields 1 = w → ((a ∨ b) → c) ≤ w → (xi → c), i.e.,
w → (xi → c) = 1. This proves (1.2.5).
In conclusion, hai ∈ f (a) and hbi ∈ f (b), and so hai u hbi ∈ f (a) ∨ f (b) which along with (1.2.5) entails hx1, . . . , xni ∈ f (a) ∨ f (b), and therefore f (a ∨ b) ⊆ f (a) ∨ f (b). The
other inclusion easily follows from a, b ≤ a ∨ b.
Remark 1.2.5 It should be emphasized here that pseudo-BCK-meet-semilattices (re-spectively, pseudo-BCK-lattices) are not the corresponding subreducts of integral resi-duated lattices (see the example below). As a matter of fact, a pseudo-BCK-meet-semilattice (A, ∧, →, , 1) can be embedded into the {∧, →, , 1}-reduct of an integral residuated lattice if and only if it satisfies the identity
x → (y ∧ z) = (x → y) ∧ (x → z), (1.2.6)
which is easily seen to hold in residuated lattices. Indeed, if (A, ∧, →, , 1) satisfies the above identity then, for all a, b ∈ A and hx1, . . . , xni ∈ P ,
hx1, . . . , xni ∈ f (a ∧ b) ⇔ xi→ (a ∧ b) = 1 for all i = 1, . . . , n,
⇔ (xi → a) ∧ (xi → b) = 1 for all i = 1, . . . , n,
⇔ xi→ a = 1 and xi → b = 1 for all i = 1, . . . , n,
⇔ hx1, . . . , xni ∈ f (a) ∩ f (b).
Example 1.2.6 The pseudo-BCK-lattice (A, ∨, ∧, →, , 1) from Fig. 1.2.1 (cf. [68]) where the operations → and are given by
→ 0 a b 1 0 1 1 1 1 a b 1 b 1 b 0 a 1 1 1 0 a b 1 0 a b 1 0 1 1 1 1 a 0 1 b 1 b a a 1 1 1 0 a b 1
does not satisfy (1.2.6)—for instance, b → (a ∧ b) = 0 6= a = (b → a) ∧ (b → b). Hence, although (A, ∨, →, , 1) is a {∨, →, , 1}-subreduct of some integral residuated lattice, the embedding cannot preserve ∧.
0 a 1 b Figure 1.2.1
1.3
Pseudo-BCK-logic
One of the convincing arguments for treating BCK-algebras as algebras (A, →, 1) rather than (A, ∗, 0) is that the approach dual to the original one makes obvious the connection with Meredith’s BCK-logic—the propositional logic with the axioms
(B) (ϕ → ψ) → ((ψ → ϑ) → (ϕ → ϑ)), (C) (ϕ → (ψ → ϑ)) → (ψ → (ϕ → ϑ)), (K) ϕ → (ψ → ϕ),
and modus ponens as the only infrence rule.
In this short section we present a non-commutative version of BCK-logic, pseudo-BCK-logic psBCK, which turns out to stand to pseudo-BCK-algebras as pseudo-BCK-logic stands to BCK-algebras, i.e., algebras are the models of the pseudo-BCK-logic. Our definition is motivated by H´ajek’s definition of a non-commutative extension of his basic logic (see [33]):
Definition 1.3.1 The formulas of the pseudo-BCK-logic (psBCK for short) are built from propositional variables and the primitive connectives → and . The axioms are the following formulas:
(B1) (ϕ → ψ) → ((ψ → ϑ) (ϕ → ϑ)), (B2) (ϕ ψ) → ((ψ ϑ) → (ϕ ϑ)), (C1) (ϕ → (ψ ϑ)) → (ψ (ϕ → ϑ)), (C2) (ϕ (ψ → ϑ)) → (ψ → (ϕ ϑ)), (K1) ϕ → (ψ → ϕ), (K2) ϕ → (ψ ϕ). The inference rules are:
1.3. Pseudo-BCK-logic 13 (Imp1) h{ϕ → ψ}, ϕ ψi, i.e., from ϕ → ψ we infer ϕ ψ,
(Imp2) h{ϕ ψ}, ϕ → ψi, i.e., from ϕ ψ we infer ϕ → ψ.
We first observe that psBCK is an algebraizable logic in the sense of [2]:
Proposition 1.3.2 The pseudo-BCK-logic is algebraizable with the set of equivalence formulas ∆ = {x → y, y → x} and the defining equation x = x → x.
Proof: Following the notation of [2], given arbitrary formulas ϕ, ψ, we shall write ϕ∆ψ as an abbreviation of {ϕ → ψ, ψ → ϕ}. In order to show that psBCK is alge-braizable, by [2], Theorem 4.7, we must check the following properties, for all formulas ϕ, ψ, ϑ, ϕ1, ψ1: 3 (i) ` ϕ∆ϕ, (ii) ϕ∆ψ ` ψ∆ϕ, (iii) ϕ∆ψ, ψ∆ϑ ` ϕ∆ϑ, (iv) ϕ∆ψ, ϕ1∆ψ1 ` (ϕ → ϕ1)∆(ψ → ψ1), (ϕ ϕ1)∆(ψ ψ1), (v) ϕ a` ϕ∆(ϕ → ϕ). We have ` ϕ → ((ϕ → (ϕ → ϕ)) ϕ) by (K2), ` (ϕ → (ϕ → ϕ)) (ϕ → ϕ) by (C1), whence ` ϕ → ϕ using Imp2, (K1) and MP. Since ϕ∆ϕ = {ϕ → ϕ}, the condition (i) is settled.
(ii) is trivial, because ϕ∆ψ = ψ∆ϕ.
Recalling (B1), ϕ∆ψ ` (ψ → ϑ) (ϕ → ϑ), so ϕ∆ψ, ψ∆ϑ ` ϕ → ϑ. By replacing ϕ and ϑ we obtain ϕ∆ψ, ψ∆ϑ ` ϑ → ϕ, which proves (iii).
By (B1) and Imp2 we have ϕ∆ψ ` (ϕ → ϕ1) → (ψ → ϕ1), thus
ϕ∆ψ ` (ϕ → ϕ1)∆(ψ → ϕ1) (1.3.1)
by symmetry. By Imp1, ϕ∆ψ ` ψ ϕ, hence ϕ∆ψ ` (ϕ ϕ1) → (ψ ϕ1) by (B2), and we have
ϕ∆ψ ` (ϕ ϕ1)∆(ψ ϕ1). (1.3.2)
Further, from (B1) and (C1) we get ` (ϕ1 → ψ1) ((ψ → ϕ1) → (ψ → ψ1)), so
ϕ1∆ψ1 ` (ψ → ϕ1)∆(ψ → ψ1). (1.3.3)
Analogously, by (B2) and (C2) we have ` (ϕ1 ψ1) → ((ψ ϕ1) (ψ ψ1)), which along with ϕ1∆ψ1 ` ϕ1 ψ1 yields ϕ1∆ψ1 ` (ψ ϕ1) (ψ ψ1), and so ϕ1∆ψ1` (ψ ϕ1) → (ψ ψ1). Hence
ϕ1∆ψ1 ` (ψ ϕ1)∆(ψ ψ1). (1.3.4)
3We write briefly ` instead of `
Now, by virtue of the property (iii), from (1.3.1) and (1.3.3) it follows ϕ∆ψ, ϕ1∆ψ1 ` (ϕ → ϕ1)∆(ψ → ψ1),
and similarly (1.3.2) and (1.3.4) yield
ϕ∆ψ, ϕ1∆ψ1 ` (ϕ ϕ1)∆(ψ ψ1). We have just proved (iv).
There remains to prove (v). Trivially, ϕ ` ϕ → (ϕ → ϕ). By (i) and Imp1 ` (ϕ → ϕ) (ϕ → ϕ), whence ` ϕ → ((ϕ → ϕ) ϕ), so that ϕ ` (ϕ → ϕ) ϕ and ϕ ` (ϕ → ϕ) → ϕ. Thus ϕ ` ϕ∆(ϕ → ϕ).
Conversely, by (i) we have ` ((ϕ → ϕ) → ϕ) ((ϕ → ϕ) → ϕ), which implies ` (ϕ → ϕ) → (((ϕ → ϕ) → ϕ) ϕ). Since ` ϕ → ϕ, it follows ` ((ϕ → ϕ) → ϕ) ϕ and ` ((ϕ → ϕ) → ϕ) → ϕ. Therefore ϕ∆(ϕ → ϕ) ` ϕ. The proof is complete.
Let us recall from [2] that the equivalent quasivariety semantics for the logic psBCK is a quasivariety K of algebras (A, →, ) of type h2, 2i satisfying certain identities and quasi-identities, which are derived from the axioms and inference rules of psBCK using ∆ = {x → y, y → x} and x = x → x, such that
(i) for every set of formulas Σ and every formula ϕ,
Σ `psBCK ϕ iff {ψ = ψ → ψ : ψ ∈ Σ} |=K ϕ = ϕ → ϕ,
(ii) for every formulas ϕ, ψ,
ϕ = ψ =||=K {ϕ → ψ = (ϕ → ψ) → (ϕ → ψ), ψ → ϕ = (ψ → ϕ) → (ψ → ϕ)}.
Observe that |=K ϕ → ψ = (ϕ → ψ) → (ϕ → ψ) iff `psBCK ϕ → ψ, and likewise
|=K ψ → ϕ = (ψ → ϕ) → (ψ → ϕ) iff `psBCKψ → ϕ. Hence
|=K ϕ = ψ iff (`psBCKϕ → ψ & `psBCKψ → ϕ) iff `psBCK ϕ∆ψ. We have the following analogue of Theorem 5.11 in [2]:
Theorem 1.3.3 The quasivariety of pseudo-BCK-algebras is (termwise equivalent to) the equivalent quasivariety semantics for the pseudo-BCK-logic.
Proof: We first note that ` ϕ → ((ψ → ψ) ϕ), whence ` (ψ → ψ) (ϕ → ϕ) and ` (ψ → ψ) → (ϕ → ϕ). Thus ` (ϕ → ϕ)∆(ψ → ψ) by symmetry. It can be analogously shown that ` (ϕ → ϕ)∆(ϕ ϕ) and ` (ϕ ϕ)∆(ψ ψ), which means that the equivalent algebraic semantics K satisfies the identities x → x = y → y = y y. Therefore every algebra (A, →, ) in K possesses a constant 1 where 1 = a → a = a a for all a ∈ A. Let K∗ be the class consisting of algebras
1.3. Pseudo-BCK-logic 15 (A, →, , 1) such that (A, →, ) belongs to K . By [2], Theorem 2.17, the quasivariety K∗ is axiomatized as follows: (x → y) → ((y → z) (x → z)) = 1, (1.3.5) (x y) → ((y z) → (x z)) = 1, (1.3.6) (x → (y z)) → (y (x → z)) = 1, (1.3.7) (y (x → z)) → (x → (y z)) = 1, (1.3.8) x → (y → x) = 1, (1.3.9) x → (y x) = 1, (1.3.10) x → x = 1, (1.3.11) x = 1 & x → y = 1 ⇒ y = 1, (1.3.12) x → y = 1 ⇒ x y = 1, (1.3.13) x y = 1 ⇒ x → y = 1, (1.3.14) x → y = 1 & y → x = 1 ⇒ x = y. (1.3.15)
It is obvious that every pseudo-BCK-algebra satisfies (1.3.5)—(1.3.15), so the qua-sivariety of pseudo-BCK-algebras is included in K∗.
Conversely, let (A, →, , 1) be an algebra belonging to K∗. By Theorem 1.1.10 we
have to show that it satisfies the equations
1 → x = x, 1 x = x and x → 1 = 1. From (1.3.7), (1.3.8) and (1.3.15) it follows that the identity
x → (y z) = y (x → z)
holds in K∗. Hence 1 → ((1 → x) x) = (1 → x) (1 → x) = 1 using (1.3.11) and (1.3.13), whence (1 → x) x = 1 by (1.3.12), and so (1 → x) → x = 1 by (1.3.14). But by (1.3.9) we also have x → (1 → x) = 1, and consequently, by (1.3.15) we obtain x = 1 → x. The argument for x = 1 x is similar, and the identity x → 1 = 1 follows from (1.3.9) and (1.3.11) since x → 1 = x → (x → x) = 1.
Deductive systems and
congruences
In this chapter, we study the lattice of deductive systems and its sublattice of com-patible deductive systems, which correspond to the relative congruences of a pseudo-BCK-algebra. In Section 2.3, we prove that representable integral residuated lattices and pseudo-BCK-algebras can be characterized by the identity
((x y) u) (([([((y x) z) z] → w) → w] u) u) = 1.
2.1
The lattice of deductive systems
The concept of a deductive system was introduced in [35] as a natural generalization of convex subalgebras (or prefilters in [67]) of integral residuated lattices (cf. [4], [46]). From the logical point of view, deductive systems correspond to those sets of formulas which are closed under the inference rule modus ponens (MP).
Definition 2.1.1 Let (A, →, , 1) be a pseudo-BCK-algebra. We say that D ⊆ A is a deductive system of A if
(DS1) 1 ∈ D,
(DS2) for all a, b ∈ A, if a ∈ D and a → b ∈ D, then b ∈ D.
It is not hard to show that the deductive systems of the reduct (L, →, , 1) of an integral residuated lattice (L, ∨, ∧, ·, →, , 1) can be characterized as non-empty subsets which are closed under · and are order-filters. The terminology in literature is a bit confusing, because several different names are used for such subsets: for instance, prefilters in case of integral residuated lattices [67], and filters in case of pseudo-BL-algebras [11].
In a sense, deductive systems can also be regarded as an analogue of convex `-sub-groups of lattice-ordered `-sub-groups, since D ⊆ G−is a deductive system of (G−, →, , 1)
iff D = G−∩ K for some convex `-subgroup K (actually, K is the convex `-subgroup
generated by D) of an `-group (G, ·,−1, 1, ∨, ∧).
Some of the results we present here and in the next section were independently obtained in [68] and [53] (our manuscript [35] has been submitted in 2005).
2.1. The lattice of deductive systems 17 Lemma 2.1.2 Every deductive system of (A, →, , 1) is an order-filter of (A, ≤).
Proof: Let a ∈ D and a ≤ b ∈ A. Then a → b = 1 ∈ D whence b ∈ D. The condition (DS2) can be equivalently formulated with in place of →:
Lemma 2.1.3 Let A be a pseudo-BCK-algebra. Then D ⊆ A with 1 ∈ D is a deductive system if and only if it satisfies the condition
(DS2’) for all a, b ∈ A, if a ∈ D and a b ∈ D, then b ∈ D.
Proof: Assume that D ⊆ A is a deductive system. If a ∈ D and a b ∈ D then a ≤ (a b) → b implies (a b) → b ∈ D and hence b ∈ D. Conversely, if D satisfies (DS2’) then the inequality a ≤ (a → b) b analogously entails b ∈ D whenever a ∈ D, a → b ∈ D.
The set DS(A) of all deductive systems of a pseudo-BCK-algebra (A, →, , 1), partially ordered by set-inclusion, is obviously a complete lattice where infima coincide with set-theoretical intersections. Hence for every X ⊆ A there exists the smallest deductive system containing X; it is denoted by D(X) and called the deductive system generated by X. We write D(x1, . . . , xn) for D(X) when X = {x1, . . . , xn}. For every
x ∈ A, D(x) is the principal deductive system generated by x.
For convenience, we shall abbreviate the terms x → (· · · → (x → y) · · · ) and x (· · · (x y) · · · ) by x →n y and x n y, n ∈ N
0 indicating the number of occurrences of x. More precisely, for x, y ∈ A and n ∈ N0 we define x →ny inductively as follows:
x →0y = y,
x →ny = x → (x →n−1 y) for n ≥ 1; x ny is defined in the same way.
Lemma 2.1.4 Let (A, →, , 1) be a psedudo-BCK-algebra. Then D(∅) = {1}, and for every ∅ 6= X ⊆ A we have
D(X) = {a ∈ A : x1 → (· · · → (xn→ a) · · · ) = 1 for some x1, . . . , xn∈ X, n ∈ N}.
In particular, for every x ∈ A,
D(x) = {a ∈ A : x →na = 1 for some n ∈ N}.
Proof: Trivially, D(∅) = {1}. Let X be a non-empty subset and let M denote the right-hand side above. It is clear that 1 ∈ M . To see that M is a deductive system assume a ∈ M and a b ∈ M , i.e., x1 → (· · · → (xm → (a b)) · · · ) = 1 and y1→ (· · · → (yn→ a) · · · ) = 1 for some x1, . . . , xm, y1, . . . , yn∈ X, m, n ∈ N.
Then a (x1 → (· · · → (xm → b) · · · )) = x → (· · · → (xm → (a b)) · · · ) = 1,
thus a ≤ x1 → (· · · → (xm → b) · · · ), which yields 1 = y1 → (· · · → (yn → a) · · · ) ≤ y1→ (· · · → (yn→ (x1 → (· · · → (xm → b) · · · ))) · · · ). Since all yi’s and xj’s belong to
X, it follows that b ∈ M . We have proved M ∈ DS(A).
Moreover, one readily sees that (i) X ⊆ M , and (ii) M ⊆ D whenever X ⊆ D for D ∈ DS(A), so that M = D(X) as desired.
It is obvious that in the formulas for D(X) and D(x) we may alternatively use the operation , that is, for every ∅ 6= X ⊆ A and x ∈ A we have
D(X) = {a ∈ A : x1 (· · · (xn a) · · · ) = 1 for some x1, . . . , xn∈ X, n ∈ N}
and
D(x) = {a ∈ A : x na = 1 for some n ∈ N}.
Corollary 2.1.5 For any pseudo-BCK-algebra (A, →, , 1), DS(A) forms an algebraic lattice whose compact elements are precisely the finitely generated deductive systems.
Proof: It easily follows from the prevoius lemma that the mapping X 7→ D(X) is an algebraic closure operator on the power set of A, i.e., for every X ⊆ A,
D(X) =[{D(X0) : X0 is a finite subset of X}, and hence DS(A) is an algebraic lattice.
In contrast to integral residuated lattices, not all compact deductive systems are principal and, more significantly, the intersection of two principal deductive systems need not be compact, thus, in general, Com(DS(A)) is not a sublattice of DS(A): Example 2.1.6 Let (A, →, 1) be the BCK-algebra associated to the poset ({1, a, b} ∪ C, ≤), where 1 is the top element, C is an infinite set of coatoms, a k b and a, b < c < 1 for all c ∈ C; that is,
x → y = (
1 for x ≤ y, y otherwise.
It is easily seen that D(a) = {1, a} ∪ C and D(b) = {1, b} ∪ C, hence D(a) ∩ D(b) = {1} ∪ C. But for every finite C0 ⊆ C we have D(C0) = {1} ∪ C0, thus D(a) ∩ D(b) is not finitely generated.
The intersection of two principal deductive system is given as follows:
Proposition 2.1.7 Let (A, →, , 1) be a pseudo-BCK-algebra, and x, y ∈ A. Then D(x) ∩ D(y) = D(U (x, y)),
where U (x, y) = {a ∈ A : a ≥ x and a ≥ y}.
Proof: We first prove that D(U (x, y)) ⊆ D(x) ∩ D(y). Let a ∈ D(U (x, y)), i.e., z1 → (· · · → (zn → a) · · · ) = 1 for some z1, . . . , zn ∈ U (x, y). Since x, y ≤ zi for all
i = 1, . . . , n, it follows (by an easy induction) that z1→ (· · · → (zn→ a) · · · ) ≤ x →na,
y →na. Hence x →na = y →na = 1 and so a ∈ D(x) ∩ D(y).
For the converse inclusion, let a ∈ D(x) ∩ D(y), i.e., x →n a = 1 and y →n a = 1
for some n ∈ N. Proceeding inductively, we prove that there exist z1, . . . , zq∈ U (x, y),
q ∈ N, such that zq→ (· · · → (z1 → a) · · · ) = 1, which means a ∈ D(U (x, y)).
For n = 1 we have x → a = y → a = 1, thus a ∈ U (x, y) and we may take z1 = a. Suppose that the statement holds for all positive integers k ≤ n. Let x →n+1 a = y →n+1a = 1. From y →n+1a = 1 we obtain y → (y na) = 1, hence
2.1. The lattice of deductive systems 19 From y na ≥ a it follows x n+1(y na) ≥ x n+1a = 1, so x n+1 (y na) = 1, which yields
x → (x n(y na)) = 1. (2.1.2)
Now, by the first induction step, and by (2.1.1) and (2.1.2), we conclude that 1 = z1 → (x n(y na)) = x n(y n(z1→ a))
for some z1 ∈ U (x, y), and therefore,
x → (x n−1(y n(z1 → a))) = 1. (2.1.3) Further, z1 → a ≥ a entails y →n+1(z
1 → a) ≥ y →n+1a = 1, so y →n+1 (z1 → a) = 1 and y → (y n(z1 → a)) = 1. Hence
y → (x n−1(y n(z1 → a))) =
= x n−1(y → (y n(z1 → a))) = x n−11 = 1. (2.1.4) By the first induction step, from (2.1.3) and (2.1.4) we get z2 ∈ U (x, y) such that
1 = z2 → (x n−1 (y n (z1 → a))) = x n−1 (y n (z2 → (z1 → a))), whence x → (x n−2(y n(z2→ (z1 → a)))) = 1. (2.1.5) Analogously, z2→ (z1 → a) ≥ a implies y →n+1(z 2 → (z1→ a)) ≥ y →n+1a = 1, and consequently, y → (y n(z
2→ (z1 → a))) = 1. This yields y → (x n−2(y n(z2 → (z1→ a)))) =
= x n−2(y → (y n(z2→ (z1 → a)))) = x n−21 = 1. (2.1.6) By (2.1.5) and (2.1.6), there is z3 ∈ U (x, y) such that
1 = z3 → (x n−2(y n(z2 → (z1 → a)))) =
= x n−2(y n(z3 → (z2 → (z1 → a)))). Repeating this procedure we obtain
y n(zn+1→ (· · · → (z1 → a) · · · )) = 1 for some z1, . . . , zn+1∈ U (x, y), and equivalently,
y →n(zn+1→ (· · · → (z1 → a) · · · )) = 1. (2.1.7) When interchanging x and y, we have
x →n(zn+1→ (· · · → (z1 → a) · · · )) = 1. (2.1.8) Now, applying the induction hypothesis to (2.1.7) and (2.1.8), there exist zn+2, . . . , zq∈
U (x, y) such that zq → (· · · → (zn+2 → [zn+1 → (· · · → (z1 → a) · · · )]) · · · ) = 1. The proof is complete.
In what follows, we prove that the lattice of deductive systems DS(A) is relatively pseudocomplemented, hence (join-infinitely) distributive. We start by introducing a concept that is intimately related to pseudocomplementation in DS(A).
Let (A, →, , 1) be a pseudo-BCK-algebra, ∅ 6= X ⊆ A. The set hXir= {a ∈ A : a → x = x for all x ∈ X}
is called the right annihilator of X (see [35]1), and the set hXil= {a ∈ A : a x = x for all x ∈ X} is called the left annihilator of X. It should be noticed that
hXir= {a ∈ A : (a → x) x = 1 for all x ∈ X}
and
hXil = {a ∈ A : (a x) → x = 1 for all x ∈ X}.
When X = {x1, . . . , xn} we shall simply write hx1, . . . , xnir and hx1, . . . , xnil for hXir
and hXil, respectively.
Lemma 2.1.8 Let (A, →, , 1) be a pseudo-BCK-algebra. For any ∅ 6= X ⊆ A, the annihilators hXir and hXil are deductive systems. In addition, if X is an order-filter
of (A, ≤) then hXir= hXil.
Proof: For every x ∈ X we have hxir = {a ∈ A : a → x = x}. Clearly, 1 ∈ hxir.
If a, a → b ∈ hxir, then x ≤ b → x ≤ (a → b) → (a → x) = (a → b) → x = x, so b → x = x showing b ∈ hxir. Thus hxir ∈ DS(A). Since hXir =
T
{hxir : x ∈ X},
we conclude that hXir ∈ DS(A). It can be analogously proved that also hXil is a
deductive system.
Let X be an order-filter. For every a ∈ hXir and x ∈ X we have (a x) → x ∈ X
for (a x) → x ≥ x ∈ X. Hence 1 = a → ((a x) → x) = (a x) → x, so that a ∈ hXil. Similarly, hXil⊆ hXir.
0 a 1 c b Figure 2.1.1
If X is not an order-filter it may well happen that hXir6= hXil:
2.1. The lattice of deductive systems 21 Example 2.1.9 Let (A, →, , 1) be a pseudo-BCK-algebra from Fig. 2.1.1 (cf. [68]) with the operations →, given as follows:
→ 0 a b c 1 0 1 1 1 1 1 a b 1 1 1 1 b 0 c 1 c 1 c b b b 1 1 1 0 a b c 1 0 a b c 1 0 1 1 1 1 1 a c 1 1 1 1 b c c 1 c 1 c 0 b b 1 1 1 0 a b c 1
For instance, we have h0ir= {b, 1} while h0il = {c, 1}.
It is easily seen that X ⊆ Y implies hY ir ⊆ hXir as well as hY il⊆ hXil. However,
other expected properties fail to be true; specifically, the “double annihilators” hhXirir,
hhXiril, hhXilir and hhXilil need not exceed X. Indeed, in Example 2.1.9 we have
hh0irir= hh0iril= {c, 1} and hh0ilir= hh0ilil= {b, 1}.
Proposition 2.1.10 Let (A, →, , 1) be a pseudo-BCK-algebra and D ∈ DS(A). Then hDir = hDil is the pseudocomplement of D in the lattice DS(A).
Proof: Let D ∈ DS(A). Since D is an order-filter, it follows hDir= hDil. We have
to show that E ⊆ hDir iff E ∩ D = {1}, for every E ∈ DS(A). If E ⊆ hDir then
a = a → a = 1 for each a ∈ E ∩ D, so E ∩ D = {1}. Conversely, if E ∩ D = {1} then for any e ∈ E and d ∈ D, (e → d) d ≥ e, d entails (e → d) d = 1, and therefore e → d = ((e → d) d) → d = 1 → d = d, so e ∈ hDir yielding E ⊆ hDir.
As we have promised, we now describe the relative pseudocomplements in the de-ductive system lattice of (A, →, , 1): Given D, E ∈ DS(A), we define the relative annihilator of D with respect to E to be the set
hD, Ei = {a ∈ A : (a → d) d ∈ E for all d ∈ D}. We should observe that
hD, Ei = {a ∈ A : (a d) → d ∈ E for all d ∈ D}.
Indeed, if a ∈ hD, Ei then for arbitrary d ∈ D we have (a d) → d ∈ D and hence (a d) → d = 1 ((a d) → d) = [a → ((a d) → d)] ((a d) → d) ∈ E. By interchanging the two arrows we obtain the converse implication.
Proposition 2.1.11 Let (A, →, , 1) be a pseudo-BCK-algebra and D, E ∈ DS(A). Then the relative annihilator hD, Ei is the relative pseudocomplement of D with respect to E in the lattice DS(A).
Proof: First we prove that hD, Ei is a deductive system. Obviously, 1 ∈ hD, Ei. Suppose a, a → b ∈ hD, Ei. For every x ∈ D, since a → x ∈ D, it follows ((a → b) → (a → x)) (a → x) ∈ E. Further, b → x ≤ (a → b) → (a → x) implies (b → x) (a → x) ≥ ((a → b) → (a → x)) (a → x) ∈ E, whence (b → x) (a → x) ∈ E, and finally, ((b → x) (a → x)) ((b → x) x) ≥ (a → x) x ∈ E yields
((b → x) (a → x)) ((b → x) x) ∈ E whence we get (b → x) x ∈ E. Thus b ∈ hD, Ei, and hence hD, Ei ∈ DS(A).
It remains to show that C ⊆ hD, Ei if and only if C ∩ D ⊆ E for each C ∈ DS(A). Let C ⊆ hD, Ei. If a ∈ C ∩ D then a = (a → a) a ∈ E, so C ∩ D ⊆ E. Conversely, let C ∩D ⊆ E and let a ∈ C. Then for every x ∈ D we have (a → x) x ∈ C ∩D ⊆ E. Thus a ∈ hD, Ei and C ⊆ hD, Ei.
We have already seen that DS(A) is an algebraic lattice. Therefore, since it is rela-tively pseudocomplemented, it follows that DS(A) satisfies the join-infinite distributive law C ∩_ i∈I Di= _ i∈I (C ∩ Di).
Corollary 2.1.12 For every pseudo BCK-algebra (A, →, , 1), DS(A) is an algebraic distributive lattice.
In accordance with Proposition 2.1.10 we can simply write hDi instead of hDir
or hDil and say that hDi is the annihilator of D provided D ∈ DS(A). Note that hDi = hD, {1}i. Since the lattice DS(A) is join-infinitely distributive and the pseudo-complement of D ∈ DS(A) is precisely the annihilator hDi, by the well-known Glivenko-Frink theorem we obtain:
Corollary 2.1.13 For any pseudo-BCK-algebra (A, →, , 1), the annihilators of de-ductive systems form a complete boolean lattice under inclusion, where infima agree with set-theoretical intersections and, for every familly {Pi : i ∈ I} of annihilators of
deductive systems, the supremum is given by hTi∈IhPiii.
The phrase “annihilators of deductive systems” is used in order to underline that the theorem does not hold for right or left annihilators of general subsets (because, roughly speaking, some annihilators are not annihilators of deductive systems) as shown in the following simple example:
Example 2.1.14 The set A = {0, a, b, 1} equipped with the operations →, given by the following tables is a proper pseudo-BCK-algebra where 0 < a < b < 1:
→ 0 a b 1 0 1 1 1 1 a a 1 1 1 b a a 1 1 1 0 a b 1 0 a b 1 0 1 1 1 1 a b 1 1 1 b 0 a 1 1 1 0 a b 1
The deductive systems are {0, a, b, 1} = D(0) = D(a), {b, 1} = D(b) and {1} = D(1). We have hair = hail = {b, 1} (but {b, 1} doesn’t rise as the annihilator of any deductive
system), thus every deductive system is the right/left annihilator of some subset of A, so the right/left annihilators form a lattice which coincides with DS(A) and is not boolean (it is a three-element chain).
2.1. The lattice of deductive systems 23 As a final remark we mention that X ⊆ D(X) entails hD(X)i ⊆ hXir, hXil.
How-ever, in the previous example we have hair= hail= {b, 1} whereas hD(a)i = hAi = {1}.
The question under which conditions the right annihilator of a subset X equals the an-nihilator of the deductive system D(X) was resolved in [35]:
Proposition 2.1.15 In any pseudo-BCK-algebra (A, →, , 1), the following state-ments are equivalent:
(i) hXir= hD(X)i for every ∅ 6= X ⊆ A;
(ii) for every x, y ∈ A, x → y = y iff y → x = x.
Proof: (i) ⇒ (ii). Let y → x = x, i.e. y ∈ hxir = hD(x)i. From (x → y) y ≥ x it follows (x → y) y ∈ D(x), and hence 1 = y → ((x → y) y) = (x → y) y which entails y = 1 → y = ((x → y) y) → y = x → y.
(ii) ⇒ (i). First, we note that the satisfaction of (ii) yields X ⊆ hhXirir for each ∅ 6= X ⊆ A. Indeed, if x ∈ X then for every a ∈ hXir, a → x = x which is equivalent
to x → a = a, so x ∈ hhXii.
Now, let a ∈ hXir. Then X ⊆ hhXirir ⊆ hair, whence D(X) ⊆ hair, and
conse-quently, {a} ⊆ hhairir⊆ hD(X)i, which yields hXir ⊆ hD(X)i. The other inclusion is obvious since X ⊆ D(X).
Proposition 2.1.7 allows one to characterize the meet-prime elements of DS(A), which will be referred to as the prime deductive systems of (A, →, , 1). Thus P ∈ DS(A) is prime iff, for all C, D ∈ DS(A), whenever C ∩ D ⊆ P then C ⊆ P or D ⊆ P . Since DS(A) is an algebraic distributive lattice, it follows that meet-primeness coincides with meet-irreducibility, and therefore every D ∈ DS(A) is obtained as an intersection of prime deductive systems (because D equals the intersection of the completely meet-irreducible deductive systems exceeding D).
Proposition 2.1.16 Given a pseudo-BCK-algebra (A, →, , 1), for every P ∈ DS(A), the following statements are equivalent:
(i) P is a prime deductive system,
(ii) for all x, y ∈ A, if U (x, y) ⊆ P then x ∈ P or y ∈ P .
Proof: (i) ⇒ (ii). Recalling Proposition 2.1.7, if U (x, y) ⊆ P then D(x) ∩ D(y) = D(U (x, y)) ⊆ P , whence D(x) ⊆ P or D(y) ⊆ P , so x ∈ P or y ∈ P .
(ii) ⇒ (i). Suppose P = C ∩ D for C, D ∈ DS(A) \ {P }. Then U (c, d) ⊆ C ∩ D = P for any c ∈ C \ P and d ∈ D \ P , which yields c ∈ P or d ∈ P , a contradiction. Remark 2.1.17 There is a close relation between the annihilators of deductive systems and the minimal prime deductive systems: using Lemma 2.4 in [66], since the lattice DS(A) is algebraic and distributive, for each D ∈ DS(A) we have
2.2
Compatible deductive systems
As usual, by a congruence kernel in a pseudo-BCK-algebra (A, →, , 1) we mean a subset K of A such that2 K = [1]
Θ for some Θ ∈ Con(A). One readily sees that the kernel [1]Θ of any congruence Θ ∈ Con(A) is a deductive system, but if A is a proper pseudo-BCK-algebra, then not all deductive systems are obtained as congruence kernels (for instance, the pseudo-BCK-algebra from Example 2.1.14 is simple, so that {b, 1} is not a congruence kernel). Hence our next objective is to characterize the congruence kernels of pseudo BCK-algebras as certain deductive systems.
Definition 2.2.1 Let (A, →, , 1) be a pseudo-BCK-algebra. We say that K ∈ DS(A) is a compatible deductive system if, for all a, b, ∈ A,
a → b ∈ K iff a b ∈ K.
Observe that for every Θ ∈ Con(A), the kernel [1]Θ is a compatible deductive system. Indeed, if a → b ∈ [1]Θ then also ((a → b) b) b ∈ [1]Θ, and since a b ≥ ((a → b) b) b, it follows a b ∈ [1]Θ. Similarly, a b ∈ [1]Θ yields a → b ∈ [1]Θ.
Proposition 2.2.2 Let (A, →, , 1) be a pseudo-BCK-algebra, K ∈ DS(A). Then K is a compatible deductive system iff the relation ΘK defined by the stipulation
(a, b) ∈ ΘK iff a → b ∈ K and b → a ∈ K (2.2.1)
is a congruence. In this case, [1]ΘK = K.
Proof: Suppose that K is compatible. Obviously, ΘK is reflexive and symmetric. Let (a, b), (b, c) ∈ ΘK. Then c → b ∈ K and c → b ≤ (b → a) (c → a) together imply
(b → a) (c → a) ∈ K whence c → a ∈ K. Similarly, a → b ≤ (b → c) (a → c) yields a → c ∈ K, so that (a, c) ∈ ΘK proving that ΘK is an equivalence relation.
To see that ΘK is compatible with → assume that (a, b), (c, d) ∈ ΘK. Then from
c → d ≤ (b → c) → (b → d) it follows (b → c) → (b → d) ∈ K, and also d → c ≤ (b → d) → (b → c) entails (b → d) → (b → c) ∈ K, hence (b → c, b → d) ∈ ΘK. Analogously, from a → b ≤ (b → c) (a → c) and b → a ≤ (a → c) (b → c) we obtain (a → c, b → c) ∈ ΘK. Due to transitivity of ΘK we have (a → c, b → d) ∈ ΘK.
Conversely, recalling 1 → x = x and x → 1 = 1 we observe K = [1]ΘK. Thus, if
ΘK ∈ Con(A) then its kernel K is a compatible deductive system.
It is to be emphasized, albeit it comes as no surprise, that the above proposition does not furnish a one-to-one correspondence between compatible deductive systems and congruences, because two distinct congruences of a given pseudo-BCK-algebra may have the same kernel. The following example of such a pseudo-BCK-algebra is a generalization of that from [69]:
2For any binary relation Θ on A and c ∈ A, [c]