compatible deductive system of (A, →, , 1) satisfying the following additional condi- tion, for all a, b, c ∈ A:
c → a ∈ K & c → b ∈ K ⇒ c → (a ∧ b) ∈ K. (3.1.1) Then the relation ΦK defined by
(a, b) ∈ ΦK ⇔ (a → b) ∧ (b → a) ∈ K (3.1.2)
is a congruence on (A, ∧, →, , 1) with [1]ΦK = K.
Conversely, for every congruence Θ on (A, ∧, →, , 1), the kernel [1]Θ is a compat- ible deductive system satisfying (3.1.1) and we have Φ[1]Θ = Θ.
Proof: Let K be a compatible deductive system that fulfils (3.1.1). First of all, when putting c = 1 in (3.1.1), we get that K is a filter. Therefore, (a, b) ∈ ΦK iff
(a → b) ∧ (b → a) ∈ K iff a → b ∈ K and b → a ∈ K iff (a, b) ∈ ΘK (where ΘK is defined by (2.2.1)), so that ΦK = ΘK and hence, by Proposition 2.2.2, ΦK is
a congruence relation on (A, →, , 1), the kernel of which is K. It remains to show that ΦK is compatible with ∧. For that purpose, let (a, b) ∈ ΦK, c ∈ A. We have (a ∧ c) → c = 1 ∈ K and (a ∧ c) → b ∈ K since (a ∧ c) → b ≥ a → b ∈ K. By (3.1.1) this implies (a ∧ c) → (b ∧ c) ∈ K. Analogously, (b ∧ c) → (a ∧ c) ∈ K, and so (a ∧ c, b ∧ c) ∈ ΦK.
Conversely, let Θ be a congruence of (A, ∧, →, , 1). It is clear that [1]Θ is a compatible deductive system. We prove that it enjoys the property (3.1.1). Assume that (c → a, 1) ∈ Θ and (c → b, 1) ∈ Θ. Then ((c → a) a, a) ∈ Θ whence (c, c ∧ a) = (c ∧ ((c → a) a), c ∧ a) ∈ Θ, and similarly, (c, c ∧ b) ∈ Θ. Thus (c, a ∧ b ∧ c) ∈ Θ which entails (c → (a ∧ b), 1) = (c → (a ∧ b), (a ∧ b ∧ c) → (a ∧ b)) ∈ Θ. Consequently, since [1]Θ is the kernel of Θ as well as of Φ[1]Θ, it follows that Θ =
Φ[1]Θ.
The same result holds for pseudo-BCK-lattices:
Theorem 3.1.6 Let (A, ∨, ∧, →, , 1) be a pseudo-BCK-lattice. If K is a compatible deductive system satisfying the condition (3.1.1) then the relation ΦK defined by (3.1.2) is a congruence on (A, ∨, ∧, →, , 1) such that [1]ΦK = K.
Conversely, for every congruence Θ on (A, ∨, ∧, →, , 1), the kernel [1]Θ is a com- patible deductive system satisfying (3.1.1) and we have Φ[1]Θ = Θ.
3.2
More on the lattice DS(A)
It follows from Proposition 2.1.7 (and also from Lemma 2.3.4) that, in any pseudo- BCK-algebra, D(x) ∩ D(y) = {1} if x ∨ y = 1. An even stronger result holds in case of pseudo-BCK-semilattices:
Proposition 3.2.1 Let (A, ∨, →, , 1) be a pseudo-BCK-semilattice. Then D(x) ∩ D(y) = D(x ∨ y)
for all x, y ∈ A. Consequently, the join-subsemilattice Com(DS(A)) of all compact deductive systems is a sublattice of DS(A).
Proof: In view of Proposition 2.1.7 we have D(x ∨ y) ⊆ D(U (x, y)) = D(x) ∩ D(y). For the converse, assume 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 easily follows z1 → (· · · → (zn → a) · · · ) ≤ (x ∨ y) →n a. Hence (x ∨ y) →n a = 1, which entails
a ∈ D(x ∨ y).
For the latter claim, let D(X), D(Y ) ∈ Com(DS(A)), i.e., X, Y are finite subsets of A. Because DS(A) is a distributive lattice, we have
D(X) ∩ D(Y ) =_{D(x) : x ∈ X} ∩_{D(y) : y ∈ Y } =_{D(x) ∩ D(y) : x ∈ X, y ∈ Y } =_{D(x ∨ y) : x ∈ X, y ∈ Y } = D({x ∨ y : x ∈ X, y ∈ Y }), so D(X) ∩ D(Y ) is finitely generated.
Our next objective is to show that in the presence of joins the pseudocomplements in the deductive system lattice DS(A) can alternatively be characterized as the so-called polars which generally are better behaved than annihilators (cf. Proposition 2.1.10):
Given a pseudo-BCK-semilattice (A, ∨, →, , 1), by the polar of ∅ 6= X ⊆ A we mean the set
Xδ = {a ∈ A : a ∨ x = 1 for all x ∈ X}.
We write xδ instead of {x}δ. It is easily seen that Xδ =T{xδ : x ∈ X}; other obvious
properties are: (a) X ⊆ Xδδ,
(b) X ⊆ Y ⇒ Xδ ⊇ Yδ, (c) Xδδδ= Xδ.
Proposition 3.2.2 Let (A, ∨, →, , 1) be a pseudo-BCK-semilattice. For every non- empty X ⊆ A, Xδ ∈ DS(A) and Xδ = D(X)δ. In addition, if D ∈ DS(A) then
Dδ= hDi.
Proof: Take an arbitrary x ∈ X and assume that a ∈ xδ and a → b ∈ xδ. Then a → b ≤ a → (b∨x) implies 1 = (a → b)∨x ≤ (a → (b∨x))∨x, so (a → (b∨x))∨x = 1. But a → (b ∨ x) ≥ b ∨ x ≥ x, and hence a → (b ∨ x) = 1, i.e., a ≤ b ∨ x. This yields 1 = a ∨ x ≤ b ∨ x and b ∨ x = 1 proving b ∈ xδ. Thus xδ ∈ DS(A), and consequently, X =T{xδ : x ∈ X} ∈ DS(A).
Now, let a ∈ Xδ. Then aδ ⊇ Xδδ ⊇ X whence aδ ⊇ D(X) for aδ is a deductive system. It follows a ∈ aδδ ⊆ D(X)δ showing Xδ ⊆ D(X)δ. The other inclusion is a
consequence of X ⊆ D(X).
Finally, assume D is a deductive system, and let a ∈ Dδ. Then x = 1 → x =
(a ∨ x) → x = a → x for every x ∈ D, so a ∈ hDi. Conversely, if a ∈ hDi then for each x ∈ D we have a ∨ x ∈ D, hence 1 = a → (a ∨ x) = a ∨ x, so a ∈ Dδ.
3.2. More on the lattice DS(A) 41 Lemma 3.2.3 Let (A, ∨, →, , 1) be a pseudo-BCK-semilattice, ∅ 6= X ⊆ A. We have Xδ ⊆ hXi
r∩ hXil, and if X is an order-filter then Xδ = hXir= hXil.
Proof: If a ∈ Xδ then x = 1 → x = (a ∨ x) → x = a → x for every x ∈ X, thus
a ∈ hXir. Similarly, a ∈ hXil. Assume that X is an order-filter. If a ∈ hXir and x ∈ X, then a ∨ x ∈ X and so a ∨ x = a → (a ∨ x) = 1. Hence a ∈ Xδ.
In general, the three sets Xδ, hXi
rand hXilcan differ from one another: In Example
2.1.9 we have 0δ= {1}, h0i
r= {b, 1}, and h0il= {c, 1}.
Likewise the relative pseudocomplements in DS(A), i.e., the relative annihilators (see Proposition 2.1.11), can be more concisely described by means of joins:
Proposition 3.2.4 In every pseudo-BCK-semilattice (A, ∨, →, , 1), we have hD, Ei = {a ∈ A : a ∨ d ∈ E for all d ∈ D}
for all D, E ∈ DS(A).
Proof: Let a ∈ hD, Ei and d ∈ D. Then a ∨ d ∈ D and hence a ∨ d = 1 (a ∨ d) = (a → (a ∨ d)) (a ∨ d) ∈ E. Conversely, if a ∨ d ∈ E for all d ∈ D, then for each d ∈ D, (a → d) d ∈ E since (a → d) d ≥ a ∨ d ∈ E. Thus a ∈ hD, Ei.
Given any pseudo-BCK-algebra (A, →, , 1), since DS(A) is an algebraic distribu- tive lattice where each principal deductive system is compact, it follows that for every D ∈ DS(A) and a ∈ A \ D (note that a /∈ D is equivalent to D(a) * D) there exists a prime deductive system P which separates a from D, in the sense that D ⊆ P but a /∈ P . For pseudo-BCK-semilattices we have the following strengthening:
Proposition 3.2.5 Let (A, ∨, →, , 1) be a pseudo-BCK-semilattice. Let D ∈ DS(A), and I be an ideal of the semilattice (A, ∨) with D ∩ I = ∅. Then there exists a prime deductive system P ∈ DS(A) such that D ⊆ P and I ∩ P = ∅.
Proof: A routine application of Zorn’s lemma yields that the set of all deductive systems having the required properties has a maximal element, say P . Assume P = X ∩Y for X, Y ∈ DS(A)\{P }. Then I ∩X 6= ∅ and I ∩Y 6= ∅ in view of the maximality of P , so there exist x ∈ I∩X and y ∈ I∩Y whence it follows x∨y ∈ I∩X∩Y = I∩P = ∅, a contradiction. Thus P is a prime deductive system.
Proposition 3.2.6 Let (A, ∨, →, , 1) be a pseudo-BCK-semilattice and P ∈ DS(A). Then P is prime if and only if, for all x, y ∈ A,
x ∨ y ∈ P ⇒ x ∈ P or y ∈ P.
Proof: This follows from Proposition 2.1.16 to the effect that x ∨ y ∈ P if and only if U (x, y) ⊆ P .
In what follows we pay special attention to the class of pseudo-BCK-semilattices satisfying the prelinearity identities
(x → y) ∨ (y → x) = 1,
including the negative cones of `-groups, pseudo-MV-algebras, pseudo-BL-algebras and representable pseudo-BCK-semilattices. The next result is an analogue of Proposition 2.3.10:
Proposition 3.2.7 Let (A, ∨, →, , 1) be a pseudo-BCK-semilattice that satisfies (3.2.1). Then for every P ∈ DS(A), the following statements are equivalent:
(i) P is prime;
(ii) for all x, y ∈ A, if x ∨ y ∈ P then x ∈ P or y ∈ P ; (iii) for all x, y ∈ A, if x ∨ y = 1 then x ∈ P or y ∈ P ; (iv) for all x, y ∈ A, x → y ∈ P or y → x ∈ P ;
(v) for all x, y ∈ A, x y ∈ P or y x ∈ P ;
(vi) the set of all deductive systems containing P is a chain (under inclusion).
Proof: By Proposition 3.2.6 we have (i) ⇔ (ii); the rest is parallel to the proof of Proposition 2.3.10.
Furthermore, it can be easily seen that the same conclusions of Corollary 2.3.11 and 2.3.12 hold for pseudo-BCK-semilattices satisfying (3.2.1), thus, in particular:
Corollary 3.2.8 The prime deductive systems of a pseudo-BCK-semilattice that fulfils (3.2.1) form a root-system.
A lower-bounded distributive lattice is relatively normal provided its set of prime ideals is a root-system under inclusion. The name comes from topological considera- tions: a topological space is hereditarily normal (not necessarily T2) iff the lattice of its open sets is relatively normal. The class consisting of the ideal lattices of relatively normal lattices is denoted by I RN (see [65], [66]). An algebraic distributive lattice L belongs to I RN iff (i) the join-subsemilattice Com(L) of compact elements is a sublattice of L, and (ii) the meet-prime elements of L form a root-system. For instance, the lattice of all convex `-subgroups of an arbitrary `-group as well as the lattice of all `-ideals (and hence the congruence lattice) of a representable `-group are members of I RN . Likewise, by [4], the congruence lattice of every representable residuated lattice belongs to I RN .
Theorem 3.2.9 If a pseudo-BCK-semilattice (A, ∨, →, , 1) satisfies the identities (3.2.1), then the deductive system lattice DS(A) is a member of I RN .
Proof: DS(A) is an algebraic distributive lattice such that (i) the compact (finitely generated) deductive systems form a sublattice Com(DS(A)) by Proposition 3.2.1, and (ii) the set of prime deductive systems is a root-system by Corollary 3.2.8. Hence DS(A) ∈ I RN .
Proposition 3.2.10 Let (A, ∨, →, , 1) be a representable pseudo-BCK-semilattice. For every x, y ∈ A we have
3.2. More on the lattice DS(A) 43 Consequently, the set Com(DSc(A)) of all compact (= finitely generated) compatible
deductive systems is a sublattice of DSc(A).
Proof: Let (A, ∨, →, , 1) be a subdirect product of the family {(At, ∨, →, , 1) :
t ∈ T } of pseudo-BCK-semilattices with underlying linear order. Let a ∈ Dc(x)∩Dc(y),
i.e., g1 → (. . . → (gm → a) . . . ) = 1 and h1 → (. . . → (hn → a) . . . ) = 1 for some
g1, . . . , gm ∈ Γ(x) and h1, . . . , hn∈ Γ(y). We shall write ˜gi and ˜hj for the elements of
Γ(x∨y) which are the same “conjugates” of x∨y as giand hjare of x and y, respectively;
thus if, for example, gi= (γa1 ◦ · · · ◦ γaq)(x) then ˜gi = (γa1◦ · · · ◦ γaq)(x ∨ y).
For fixed t ∈ T , letting πt denote the usual projection map onto At, we either have
πt(x) ≤ πt(y) or πt(x) ≥ πt(y):
Case 1: If πt(x) ≥ πt(y) then πt(x) = πt(x ∨ y) and it is easily seen that πt(gi) = πt(˜gi)
for every i = 1, . . . , m. It follows
πt(1) = πt(g1→ (. . . → (gm → a) . . . )) = πt(g1) → (. . . → (πt(gm) → πt(a)) . . . ) = πt(˜g1) → (. . . → (πt(˜gm) → πt(a)) . . . ) whence πt(1) = πt(˜h1) → (. . . → (πt(˜hn) → [πt(˜g1) → (. . . → (πt(˜gm) → πt(a)) . . . )]) . . . ) = πt(˜h1→ (. . . → (˜hn→ [˜g1 → (. . . → (˜gm→ a) . . . )]) . . . )).
Case 2: If πt(x) ≤ πt(y) then πt(y) = πt(x ∨ y) and πt(hj) = πt(˜hj) for all j = 1, . . . , n. We have
πt(1) = πt(h1→ (. . . → (hn→ a) . . . ))
= πt(h1) → (. . . → (πt(hn) → πt(a)) . . . )
= πt(˜h1) → (. . . → (πt(˜hn) → πt(a)) . . . ) which along with πt(˜g1) → (. . . → (πt(˜gm) → πt(a)) . . . ) ≥ πt(a) implies
πt(1) = πt(˜h1) → (. . . → (πt(˜hn) → [πt(˜g1) → (. . . → (πt(˜gm) → πt(a)) . . . )]) . . . )
= πt(˜h1→ (. . . → (˜hn→ [˜g1 → (. . . → (˜gm→ a) . . . )]) . . . )).
In either case, πt(1) = πt(˜h1 → (. . . → (˜hn→ [˜g1 → (. . . → (˜gm → a) . . . )]) . . . )) for
all t ∈ T . Consequently, 1 = (˜h1 → (. . . → (˜hn → [˜g1 → (. . . → (˜gm → a) . . . )]) . . . ),
which proves a ∈ Dc(x ∨ y) since ˜gi, ˜hj ∈ Γ (x ∨ y). Thus Dc(x) ∩ Dc(y) ⊆ Dc(x ∨ y).
The reverse inclusion is obvious.
Now, let Dc(X), Dc(Y ) ∈ Com(DSc(A)), i.e., X, Y are finite subsets of A. Because DSc(A) is a distributive lattice, like in the proof of the second part of Proposition 3.2.1,
we have Dc(X) ∩ Dc(Y ) = _ {Dc(x) : x ∈ X} ∩ _ {Dc(y) : y ∈ Y } =_{Dc(x) ∩ Dc(y) : x ∈ X, y ∈ Y } =_{Dc(x ∨ y) : x ∈ X, y ∈ Y } = Dc({x ∨ y : x ∈ X, y ∈ Y }),
Theorem 3.2.11 For every representable pseudo-BCK-semilattice (A, ∨, →, , 1), the lattice DSc(A) of compatible deductive systems is a member of the class I RN .
Proof: We already know that DSc(A) is an algebraic distributive lattice such that
Com(DSc(A)) is a sublattice, so there remains to be proved that the meet-prime ele-
ments of DSc(A) form a root-system. Let P be a meet-prime element of DSc(A). Let
H, K be compatible deductive systems containing P , suppose that H * K and K + H, and take arbitrary a ∈ H \ K and b ∈ K \ H. It is readily seen that b → a ∈ H \ K and a → b ∈ K \ H. Since (A, ∨, →, , 1) is representable, we have (a → b) ∨ (b → a) = 1 and hence Dc(a → b) ∩ Dc(b → a) = Dc((a → b) ∨ (b → a)) = Dc(1) = {1} ⊆ P , which
entails Dc(a → b) ⊆ P or Dc(b → a) ⊆ P . Thus a → b ∈ P ⊆ H or b → a ∈ P ⊆ K, a
contradiction.