Quantum B-Algebras

Cent. Eur. J. Math. • 11(11) • 2013 • 1881-1899 DOI: 10.2478/s11533-013-0302-0

Central European Journal of Mathematics

Quantum B-algebras

Research Article

Wolfgang Rump1∗

1 Institute for Algebra and Number Theory, University of Stuttgart, Pfaﬀenwaldring 57, 70550 Stuttgart, Germany

Received 29 December 2012; accepted 16 January 2013

Abstract: The concept of quantale was created in 1984 to develop a framework for non-commutative spaces and quantum mechanics with a view toward non-commutative logic. The logic of quantales and its algebraic semantics man- ifests itself in a class of partially ordered algebras with a pair of implicational operations recently introduced as quantum B-algebras. Implicational algebras like pseudo-eﬀect algebras, generalized BL- or MV-algebras, partially ordered groups, pseudo-BCK algebras, residuated posets, cone algebras, etc., are quantum B-algebras, and every quantum B-algebra can be recovered from its spectrum which is a quantale. By a two-fold application of the functor “spectrum”, it is shown that quantum B-algebras have a completion which is again a quantale. Every quantale Q is a quantum B-algebra, and its spectrum is a bigger quantale which repairs the deﬁciency of the inverse residuals of Q. The connected components of a quantum B-algebra are shown to be a group, a fact that applies to normal quantum B-algebras arising in algebraic number theory, as well as to pseudo-BCI algebras and quantum BL-algebras. The logic of quantum B-algebras is shown to be complete.

MSC: 03G12, 03G27, 06F07, 06F15, 06F35, 03F52, 03G25

Keywords: Quantale • Non-commutative logic • Partially ordered group © Versita Sp. z o.o.

Dedicated to B. V. M.

Introduction

As far as implicational algebras are concerned, the algebras of the title may be introduced by inverting a vivid expression of Nguyen Tien Zung [31]: To say that everything is a quantum B-algebra would be a great exaggeration, but to say that everything is contained in a quantum B-algebra would not be too far from the truth.

∗ E-mail: [email protected]

1881 Quantum B-algebras

In [36], we have shown that in a certain sense, every quantum B-algebra X contains a greatest pseudo-BCK subalgebra × I(X), a “greatest” partially ordered subgroup X , and even a greatest GPE-subalgebra E+(X), where GPE abbreviates “generalized pseudo-effect algebra” [9–11]. The algebraic semantics of almost every logic can thus be found among the special classes of quantum B-algebras. To say it from the start, the logic of quantum B-algebras is the logic of quantales. In fact, every quantum B-algebra can be embedded (dually) into a quantale, and every quantale is a quantum B-algebra. Recall that a quantale is a complete partially ordered semigroup (Q; &) where the multiplication distributes over arbitrary joins. Quantales were introduced by Mulvey on a 1984 conference in Taormina (Sicily). The very short title “&” of his seminal paper [27] refers to the non-commutative conjunction, while the infinitary join operation interprets the logical disjunction. Originally, quantales have been invented to formalize the logic of observables in quantum physics. They became useful ∗ to describe the “spaces” of C -algebras [3, 18, 24] and non-commutative geometry [6, 28–30, 34, 35]. Combining both aspects of quantales, we associate a quantalic spectrum U(X) to any quantum B-algebra X in such a way that X embeds into the opposite U(X)op of the quantale U(X) (with inverse partial order) so that X can be recovered from U(X). In general, the opposite of a quantale is not a quantale, nor a quantum B-algebra. This touches a delicate point that ought to be clarified beforehand. By the completeness of a quantale Q, the multiplication (we henceforth write · instead of &) gives rise to a pair of binary operations → and ; satisfying

x · y 6 z ⇐⇒ x 6 y → z ⇐⇒ y 6 x ; z (0)

for all x; y; z ∈ Q. As mentioned above, every quantale is a quantum B-algebra, a partially ordered system (X; →; ;) with two binary operations →; ; and additional properties (see Definition 1.2). For example, the right-hand equivalence of (0) holds for a quantum B-algebra. The role of quantales in this context is twofold. Apart from being a special quantum B-algebra, a quantale U(X) is associated to any quantum B-algebra X as a (non-commutative) spectrum. By definition, U(X) is the set of upper sets of X, so that X can be identified with the set of principal upper sets ↑x in U(X). The delicate point is that the operations →; ; in X do not come from the residuals →; ; of the multiplication in Q = U(X). While the latter are given by

_ _ y → z = {x ∈ Q : x ·y 6 z}; x ; z = {y ∈ Q : x ·y 6 z};

there are dual operations

^ ^ y z = {x ∈ Q : x ·y > z}; x z = {y ∈ Q : x ·y > z}

derived from the multiplication in Q. However these derived operations determine the multiplication in Q. If X is a quantum B-algebra, the inverse operations and of U(Q) induce the operations → and ; in X. Furthermore, there is a one-to-one correspondence X 7→ U(X) between the isomorphism classes of quantum B-algebras and logical quantales (Theorem 1.6). The fact that the quantale U(X) is again a quantum B-algebra allows us to form the iterated embedding X,→ U(X) ,→ U(U(X)). As indicated above, the operations →; ; of U(X) as a quantum B-algebra are induced by the derived operations ; of U(U(X)), while the derived operations ; of U(X) do not ﬁt into an adjunction context (0) with the opposite partial order. However, it turns out that the secondary operations ; of U(X) are the restrictions of the residuals →; ; in U(U(X)). In other words, the pair of operations →; ; and ; in U(X) is just interchanged in U(U(X)). The lack of an adjunction context like (0) for the non-residuals ; is thus repaired in U(U(X)). In particular, the composed embedding X,→ U(X) ,→ U(U(X)) makes X into a subalgebra of a quantale. Therefore, every quantum B-algebra X has a completion Xb, the subquantale of U(U(X)) generated by X, which generalizes the Dedekind–MacNeille completion of an archimedean lattice-ordered group (cf. [2, Théorème 11.3.2]). Secondly, this implies that a quantum B-algebra can be characterized as a subreduct (and subposet) of a quantale (Theorem 2.3).

1882 W. Rump

One of the referees suggested to look at the correspondence between a quantum B-algebra X and its quantalic spectrum Q = U(X) in the opposite way: The quantale Q takes the part of an algebra while X is viewed as its underlying space, similar to the duality between a bounded distributive lattice with operators and its associated relational Priestley space (see [19, Theorems 2.2.3, 2.2.4]). To this end, the infinite join operation as well as the special structure of Q = U(X) as a logical quantale (see Definition 1.5) has to be taken into account. The quantum B-algebra X can then be identified 1 1 with the set of logical [36] quantale morphisms Q → Q( ) onto the endomorphism quantale Q( ) of the poset {0; 1} with 0 < 1. The multiplication of the quantale Q corresponds to a ternary relation in X. Note, however, that the residuals of the multiplication in U(X), as explained above, do not restrict to the binary operations of X. In this context, the relationship with canonical extensions, introduced by Jónsson and Tarski [20, 21] and further developed by Gehrke and Jónsson [13, 14], might be of interest. In [8], the canonical extension is introduced for an arbitrary poset with monotonic operations. Its construction as a MacNeille completion of an “intermediate structure” (see [15]) resembles the above mentioned completion Xb of a quantum B-algebra X. If X is a residuated poset, that is, a partially ordered semigroup with binary operations → and ; satisfying (0), the completion Xb satisfies one and a half of the two characterizing properties of the canonical extension (see Section 2). Every residuated poset is a quantum B-algebra. For example, a partially ordered group is a residuated poset with

− − x → y = yx 1; x ; y = x 1y:

We characterize partially ordered groups as quantum B-algebras X which satisfy

(x → y) ; y = (x ; y) → y = x for all x; y ∈ X (Proposition 3.6). Now assume that X is unital, that is, X contains an element u which makes U(X) × into a unital quantale. By [36, Theorem 3], the whole unit group U(X) of U(X) is contained in X. However, it will turn × × out that the unit group X = U(X) need not be the largest subgroup of X. Namely, we show that every pseudo-BCI algebra [22] is a quantum B-algebra (Proposition 4.4), and that the maximal elements of a pseudo-BCI algebra X form × a subgroup which can happen to be strictly larger than the unit group X . In Section 4, we introduce the connected components of a quantum B-algebra X. We show that they form a group C(X) − even if X does not have a unit element. There is a natural epimorphism c : X C(X) such that the unit ﬁber c 1(u) is the greatest connected subalgebra of X. In particular, a group can be regarded as a quantum B-algebra whose partial order is trivial. For a unital quantum × B-algebra X, we show that the morphism c : X C(X) admits a section into the unit group X if and only if X is a semidirect product G nY of a group G by a connected unital quantum B-algebra Y (Proposition 4.3). The unital quantum B-algebras satisfying the identity

x → x = x ; x = u arise, e.g., as lattices of fractional ideals in algebraic number ﬁelds. We show that every such quantum B-algebra X admits an endomorphism x 7→ x whose image X is a partially ordered group (Theorem 4.7). If X is a pseudo-BCI algebra, × X consists of the maximal elements of X. We show that in this case, X coincides with the unit group X if and only if ∼ X = G nY with a group G and a pseudo-BCK algebra Y (Theorem 4.10). The logic (qB) of quantum B-algebras is brieﬂy discussed in Section 5. It consists of four axioms and three inference rules x; x → y ` y; x → y ` x ; y; x ; y ` x → y:

Models of (qB) are quantum B-algebras X which are locally unital. If X has a unit element u, the true propositions are associated to the upper set X + = ↑u. In the locally unital case, the single unit element u is replaced by local units e ∈ X deﬁned by the equation e → e = e. Every element of the form el(x) = x → x or er(x) = x ; x is a local unit, and the equations u → x = u ; x = x which characterize u in the unital case, are replaced by the equations

el(x) ; x = er(x) → x = x

1883 Quantum B-algebras

for a locally unital quantum B-algebra. Logically, this implies that the true propositions x are characterized by the equivalent inequalities el(x) 6 x and er(x) 6 x. The Lindenbaum–Tarski process works for (qB) and yields the free locally unital quantum B-algebra L with countably many generators. We use L to show that the logic (qB) is complete.

1. Non-commutative logic and quantales

In this section, we brieﬂy review the logical aspects of quantales which led to the concept of quantum B-algebra [36]. Let us recall ﬁrst that in classical propositional logic, the conjunction A &B and the implication A → B are related by an adjunction A &B 6 C ⇐⇒ A 6 B → C; where 6 stands for the logical entailment. If the commutativity of & is dropped, the implication splits into a left and a right implication, according to the partial maps A 7→ A&B and B 7→ A&B. We keep → as a symbol for the right implication, and use ; for the left implication which is related to → by the equivalence

x 6 y → z ⇐⇒ y 6 x ; z: (1)

In the presence of a conjunction, now denoted by · instead of &, the equivalence (1) extends to

x 6 y → z ⇐⇒ x · y 6 z ⇐⇒ y 6 x ; z; (2)

but even if there is no conjunction, the left and right implication determine each other via (1). Thus, in a wide sense, non- commutative logic can be regarded as the theory of partially ordered algebras (X; →; ;) with two binary operations → and ; satisfying (1). We call X commutative if the operations → and ; coincide. Non-commutative algebras (X; →; ;) of that type have been studied since 1939 by Ward and Dilworth [37] (residuated lattices), Bosbach [4] (pseudo-hoops) and [5] (cone algebras, bricks), Georgescu and Iorgulescu [16, 17] (pseudo-MV algebras, pseudo BCK-algebras), Dvureˇcenskij and Vetterlein [9–11] (GPE-algebras), and Galatos and Tsinakis [12] (GBL-algebras). The case where → and ; are connected with a multiplication, is given by

Deﬁnition 1.1. A residuated poset is a partially ordered semigroup (X; ·) with two binary operations → and ; which satisfy (2).

Note that the associativity of · is equivalent to the equational version of (1):

x → (y ; z) = y ; (x → z): (3)

Indeed,

(x · y) · z 6 t ⇐⇒ x · y 6 z → t ⇐⇒ y 6 x ; (z → t) x · (y · z) 6 t ⇐⇒ y · z 6 x ; t ⇐⇒ y 6 z → (x ; t)

holds for all x; y; z; t ∈ X. Note however, that (1) and (3) are not equivalent.

Deﬁnition 1.2. A quantum B-algebra [36] is a partially ordered set X with two binary operations → and ; satisfying (1) and (3), and

y 6 z =⇒ x → y 6 x → z (4)

for all x; y; z ∈ X.

1884 W. Rump

Thus every residuated poset is a quantum B-algebra. Equation (4) has a counterpart

y 6 z =⇒ x ; y 6 x ; z (5) which holds for any quantum B-algebra. In fact, (1) gives

x 6 (x → y) ; y; x 6 (x ; y) → y; (6) and thus, y 6 z implies x 6 (x ; y) → y 6 (x ; y) → z by means of (4). Hence (1) yields x ; y 6 x ; z, which proves (5). Furthermore, every quantum B-algebra satisﬁes

y → z 6 (x → y) → (x → z) (7) y ; z 6 (x ; y) ; (x ; z) (8) which can be also written as

x → y 6 (y → z) ; (x → z) (9) x ; y 6 (y ; z) → (x ; z) (10) by means of (1). By (3), the right-hand side of (9) is equal to x → ((y → z) ; z). Whence (9) follows by (4) and (6). Conversely, it is easily seen [36, Proposition 3] that (3) in Deﬁnition 1.2 can be replaced by (7) and (8).

Deﬁnition 1.3. W A quantale Q is a partially ordered semigroup (Q; ·) with arbitrary joins A (for A ⊂ Q) so that

! ! _ _ _ _ a· ai = (a·ai); ai ·a = (ai ·a) (11) i∈I i∈I i∈I i∈I holds for all a; ai ∈ Q. A quantale Q is said to be unital if the semigroup (Q; ·) admits a unit element u.

W A quantale Q can be conceived as a non-commutative topology: Elements a ∈ Q stand for the open sets, A for ∗ the union, and a·b for the non-commutative intersection. For example, the spectrum of a C -algebra or the “space” of Penrose tilings can be viewed as a quantale [24, 30]. On the other hand, if the elements a ∈ Q of a quantale Q W are interpreted as logical propositions, the union A stands for the disjunction, while a·b features the sequential conjunction “a and then b”. W W Every quantale Q has a smallest element 0 = ∅, and a greatest element 1 = Q. The residuals are determined by (2) and can be deﬁned as

_ _ a → b = {c ∈ Q : c ·a 6 b}; a ; b = {c ∈ Q : a·c 6 b}: (12)

Since every quantale Q is a complete lattice, the inverse residuals

^ ^ a b = {x ∈ Q : x ·a > b}; a b = {x ∈ Q : a·x > b} (13) are well deﬁned, too.

1885 Quantum B-algebras

Deﬁnition 1.4. Let Q be a quantale. We call an element c ∈ Q balanced if it satisﬁes

! ! ^ ^ ^ ^ c · ai = (c ·ai); ai ·c = (ai ·c): i∈I i∈I i∈I i∈I

Equivalently, c is balanced if and only if

a·c > b ⇐⇒ a > c b; c ·a > b ⇐⇒ a > c b

hold for all a; b ∈ Q. An element c of a quantale Q is said to be supercompact if for any subset A ⊂ Q,

_ c 6 A =⇒ there exists a ∈ A such that c 6 a:

The set of supercompact elements of Q will be denoted by Qsc. The product of balanced elements is balanced. If c is balanced and d supercompact, then c d and c d are supercompact.

Deﬁnition 1.5. Q logical Q a ∈ Q We say that aW quantale is if every supercompact element of is balanced and every can be represented as a join a = C with C ⊂ Qsc.

Thus, for a logical quantale Q, the set X = Qsc of supercompact elements is closed under the operations and . For x; y ∈ X, we therefore deﬁne x → y = x y; x ; y = x y

to make X into an algebra (X; →; ;). We endow X with the partial ordering induced by the partial order of Qop. Then X becomes a quantum B-algebra. The converse is less obvious: For every quantum B-algebra X, the upper sets A ⊂ X (i.e. the subsets A with a > b ∈ A ⇒ a ∈ A) can be made into a quantale U(X) by deﬁning

A·B = {x ∈ X : there exists b ∈ B such that b → x ∈ A}: (14)

It can be shown [36] that this gives an associative multiplication which distributes over set-theoretic joins. Furthermore, the set of supercompact elements of U(X) coincides with the image of the embedding X,→ U(X) given by x 7→ ↑x, and every balanced element of U(X) is supercompact. So we have

Theorem 1.6. Up to isomorphism, there is a one-to-one correspondence X 7→ U(X) between quantum B-algebras and logical quantales.

With suitable morphisms, this correspondence extends to a categorical duality. Let f : X → Y be a morphism of quantum B-algebras, that is, a monotonic map which satisﬁes

f(x → y) = f(x) → f(y); f(x ; y) = f(x) ; f(y)

for all x; y ∈ X. Then Y → f(X) ⊂ f(X) ⇐⇒ Y ; f(X) ⊂ f(X); (15)

where Y → f(X) denotes the set of elements y → f(x) with x ∈ X and y ∈ Y , and similarly for Y ; f(X). If these equivalent conditions hold, we call the morphism f spectral. The category of quantum B-algebras with spectral

1886 W. Rump

morphisms will be denoted by qB. Similarly, we call a morphism g: Q → L of quantales logical if g respects arbitrary meets and satisﬁes g(Q) → L ⊂ g(Q); g(Q) ; L ⊂ g(Q): (In contrast to (15), these two conditions are not equivalent.) By LQuant we denote the category of logical quantales with logical morphisms. Then we have an equivalence of categories [36, Theorem 2]:

U : qBop → LQuant:

For a quantum B-algebra X, we call U(X) the enveloping quantale or the quantalic spectrum of X.

2. The case of residuated posets

Every quantale is a quantum B-algebra, but the quantalic spectrum U(Q) of a quantale Q is usually bigger than Q. The partial order of Q is opposite to that of U(Q). Moreover, U(Q) is a logical quantale. Therefore, Theorem 1.6 shows that the theory of quantales (with spectral morphisms) is part of the theory of logical quantales. One might ask whether it is more natural to redeﬁne the quantalic spectrum in such a way that a quantale Q is associated to itself rather than to the bigger quantale U(Q). Such a direct embedding is well known for residuated posets. In fact, every residuated poset X naturally embeds into a quantale L(X) consisting of the lower sets A ⊂ X (i.e. a 6 b ∈ A ⇒ a ∈ A). Thus L(X) is a complete lattice with respect to inclusion. For A; B ∈ L(X), we set

A·B = {x ∈ X : there exist a ∈ A, b ∈ B such that x 6 a·b}: (16)

The embedding X,→ L(X) is given by x 7→ ↓x, and X can be recovered as the set L(X)sc of supercompact elements (cf. Ono and Komori [32, 33]). Note that the associativity of X carries over to L(X) and thus makes L(X) into a quantale. However, the embedding X,→ L(X) fails to give a quantale if the residuals are not compatible with a product. The following example shows that even in the commutative case, L(X) need not be a quantale.

Example 2.1. Let (X; →; u) be the 4-element BCI-algebra given by the following table (cf. [23, Example 3.8]).

→ u a b c u u a b c a u u b c b u a u c c c c c u

Recall that a BCI-algebra is deﬁned by the axioms

(B) y → z 6 (x → y) → (x → z); (C) x → (y → z) = y → (x → z); (I) x → x = u; together with x 6 y 6 x =⇒ x = y; where the relation 6 (a partial order) is given by

x 6 y ⇐⇒ x → y = u:

Thus every BCI-algebra is a commutative quantum B-algebra. The algebra L(X) consists of 10 elements, L(X) = {0; a; b; c; u; ab; ac; bc; abc; 1}, where 0 = ∅, 1 = {a; b; c; u}, and ab = {a; b}, etc. The lattice structure of L(X) thus looks as follows.

1887 Quantum B-algebras

The equivalences (2) give the following table of products for the elements of X:

· u a b c u u a b c a a a 0 c b b 0 b c c c 0 0 0

which yields the multiplication in L(X) via (16). Note that the multiplication in L(X) is non-commutative although the two residuals coincide. In particular, we have (a·b)·c = 0·c = 0, but a·(b·c) = a·c = c. Thus L(X) is not a quantale.

A close inspection of Example 2.1 shows that the deﬁnition of a product in L(X) via (2) is not natural as it makes use V W V of instead of . For x; y ∈ X, we have x ·y = {c ∈ L(X): x 6 y → c}. For a residuated poset X, there are two natural embeddings into a quantale:

1. the direct embedding X,→ L(X); 2. the spectral embedding X,→ U(X):

The direct embedding does not work for pseudo-BCK algebras, although every pseudo-BCK algebra X can be embedded into the (→; ;)-subreduct of a quantale [1, 25]. On the other hand, the latter result extends to arbitrary quantum B-algebras. We obtain this as a consequence of a duality result (Proposition 2.2). Recall that for a quantum B- algebra X, the residuals → and ; of X coincide with the inverse residuals and of U(X), respectively. If X is a quantale, the opposite statement is also true.

Proposition 2.2. Let Q be a quantale. For x; y ∈ Q, the inverse residuals (13) in Q coincide with the residuals (12) in U(Q).

Proof. Let A; B ∈ U(Q) be given. By (14),

x ∈ A → B ⇐⇒ ↑x ·A ⊂ B ⇐⇒ ∀ b ∈ Q ∀ a ∈ A: a → b > x ⇒ b ∈ B ⇐⇒ ∀ b ∈ Q ∀ a ∈ A: xa 6 b ⇒ b ∈ B ⇐⇒ xA ⊂ B:

For a; b ∈ Q, this gives

x ∈ ↑a → ↑b ⇐⇒ x ·↑a ⊂ ↑b ⇐⇒ xa > b ⇐⇒ x > a b ⇐⇒ x ∈ ↑(a b):

Hence ↑(a b) = ↑a → ↑b. Similarly, ↑(a b) = ↑a ; ↑b.

1888 W. Rump

Theorem 2.3. An algebra (X; →; ;) of type (2; 2) with a binary relation 6 is a quantum B-algebra if and only if X can be embedded into a quantale Q such that the operations →; ; and the relation 6 are induced by →; ;, and 6 of Q.

Proof. By Deﬁnition 1.2, every (→; ;; 6)-subalgebra of a quantale is a quantum B-algebra. Conversely, let X be a quantum B-algebra. Consider the embeddings X,→ U(X) ,→ U(U(X)). By deﬁnition, the residuals of X coincide with the inverse residuals of U(X), and by Proposition 2.2, the inverse residuals of U(X) coincide with residuals of U(U(X)). The partial order of X is opposite to the partial order of U(X), hence equal to the partial order of U(U(X)). Thus X is a subalgebra of the quantum B-algebra U(U(X)).

Let X be a quantum B-algebra. If we interchange → and ;, keeping the partial order ﬁxed, we get a quantum B-algebra X op, the opposite of X. We say that X is commutative if X = X op. The following fact [36, Proposition 12] gives another instance for the adequacy of the spectral embedding X,→ U(X).

Proposition 2.4. A quantum B-algebra X is commutative if and only if the enveloping quantale U(X) is commutative.

As mentioned in the introduction, the embedding X,→ U(X) ,→ U(U(X)) can be used to associate a completion to any quantum B-algebra, namely, the subquantale Xb of U(U(X)) generated by X. Here we only show that Xb is closely related σ to the canonical extension of X as a partially ordered set with operations [8]. By definition, the canonical extension X of X is characterized as a complete lattice containing X as a subposet which is compact and dense [8]. To explain the terminology, define a filter of X to be a non-empty down-directed set F ∈ U(X). Similarly, an ideal of X is equivalent σ V W to a filter of X op. The compactness of X ⊂ X means that for a filter F and an ideal I, the inequality F 6 I implies σ V that F ∩ I =6 ∅. The dense property states that every element of X is a join of elements of the form F and a meet of W elements I. Now let us consider the embedding X,→ Xb. For simplicity, assume that X is a residuated poset. We prove compactness and the first half of the dense property. Any element x ∈ X gives rise to an element Vx = {A ∈ U(X): x ∈ A} of U(U(X)). Since X,→ Xb respects products, Xb consists of the elements VB = {A ∈ U(X): A ∩ B =6 ∅} of U(U(X)) with B ⊂ X. For a filter F and an ideal I of X, the intersection and join in U(U(X)), respectively, are given by

^ _ F = {A ∈ U(X): F ⊂ A}; I = {A ∈ U(X): A ∩ I =6 ∅}:

V W V If F ⊂ I, then F ∩I =6 ∅, which shows that X is compact in Xb. More generally, F ⊂ VB is equivalent to F ∩B =6 ∅ for any B ⊂ X. Hence

[ ^ [ (↑b) = {A ∈ U(X): b ∈ A} = {A ∈ U(X): A ∩ B =6 ∅} = VB; b∈B b∈B which proves half of the dense property. The second half states that for any B ⊂ X, the lower set ↓B coincides with the intersection of all ideals I ⊃ B. As ideals have to be up-directed, this does not hold in general.

3. Unital quantum B-algebras

In [36] it is shown that all of the examples mentioned in Section 1 are quantum B-algebras. Most of them are unital.

Deﬁnition 3.1. A quantum B-algebra X is unital if there is an element u ∈ X with u → x = u ; x = x for all x ∈ X.

1889 Quantum B-algebras

The unit element u is always unique. If u exists, the axioms can be written as

x ; (y → z) = y → (x ; z); y → z 6 (x → y) → (x → z)

together with x 6 y ⇐⇒ u 6 x → y ⇐⇒ u 6 x ; y: (17)

In terms of the quantale U(X), an element u ∈ X is a unit element of X if and only if ↑u is a unit element of U(X).

Example 3.2. By [36, Corollary of Proposition 13] a pseudo-BCK algebra can be characterized as a unital quantum B-algebra X for which the unit element u is the greatest element of X. Every unital quantum B-algebra X has a greatest pseudo-BCK subalgebra I(X) = {x ∈ X : x → u = x ; u = u}. We call the elements of I(X) integral.

Example 3.3. By Section 1, every residuated poset X is a quantum B-algebra. We call X unital as a residuated poset if (X; ·) is a monoid.

Proposition 3.4. A residuated poset X is unital if and only if X is unital as a quantum B-algebra.

Proof. Assume that X has a unit element u as a residuated poset. Then

x 6 u → y ⇐⇒ x ·u 6 y ⇐⇒ x 6 y

holds for all x ∈ X. Hence u → y = y. Similarly, u ; y = y. Conversely, assume that u → y = y holds for all y ∈ X. Then x ·u 6 y ⇐⇒ x 6 u → y ⇐⇒ x 6 y;

which yields x · u = x. By symmetry, this completes the proof.

Example 3.5. − − Every partially ordered group G is a residuated poset with residuals x → y = yx 1, x ; y = x 1y and unit element u = 1, hence a unital quantum B-algebra. The multiplication can be recovered from each of the residuals:

xy = (y → u) → x: (18)

For the next section, we make use of the following characterization of partially ordered groups which does not refer to a unit element.

Proposition 3.6. A quantum B-algebra X is a partially ordered group if and only if X is non-empty and satisﬁes

(x → y) ; y = (x ; y) → y = x: (19)

Proof. Clearly, equations (19) hold for every partially ordered group. Conversely, let X be a quantum B-algebra which satisﬁes (19). For any x; y; z ∈ X, we have x → y = x → ((y → z) ; z), which gives

x → y = (y → z) ; (x → z): (20)

1890 W. Rump

In particular, x = (x ; y) → y = (y → y) ; ((x ; y) → y) = (y → y) ; x, which yields x → x = ((y → y) ; x) → x = y → y. So the element u = x → x is unique and satisﬁes u ; x = x. Hence u ; u = u. By symmetry, this implies that u = x → x = x ; x is a unit element of X. We introduce a multiplication as follows. For x; y ∈ X, we deﬁne x ·y = (x → u) ; y: (21) Then u·y = (u → u) ; y = u ; y = y and x ·u = (x → u) ; u = x. Thus u·x = x ·u = x holds for all x ∈ X. By (20), we get x = (x → (y → z)) ; (y → z) = y → ((x → (y → z)) ; z) = ((x → (y → z)) ; z) → y ; (y → y):

This gives x → u = (x → (y → z)) ; z → y: Hence x ·y = (x → u) ; y = (x → (y → z)) ; z, and thus

(x ·y) → z = x → (y → z):

By (17), we have x ·y 6 z ⇔ u 6 (x ·y) → z ⇔ u 6 x → (y → z) ⇔ x 6 y → z. Hence, by symmetry, X is a residuated poset. Note that (3) implies that (X; ·) is associative. Furthermore, (x ; u)·x = ((x ; u) → u) ; x = x ; x = u and x ·(x → u) = (x → u) ; (x → u) = u. Thus any x ∈ X is left and right invertible, which shows that (X; · ) is a group − with x 1 = x → u = x ; u. To show that X is a partially ordered group, assume that x 6 y. Then

x 6 (y → u) ; u = (y → u) ; (z → z) = z → ((y → u) ; z) = z → (y·z):

− − Hence x ·z 6 y·z. Similarly, z 1 ·z ·x 6 y gives z ·x 6 z 1 ; y = z ·y. Finally, (20) shows that the multiplication (21) coincides with (18).

Deﬁnition 3.7. Let X be a unital quantum B-algebra. We call an element a ∈ X invertible if the equations

(a → u) → (a → x) = x; (a ; u) ; (a ; x) = x (22) hold for all x ∈ X.

× By [36, Theorem 3], the invertible elements form a subalgebra X which is a partially ordered group, and the embedding × ∼ × X,→ U(X) restricts to a group isomorphism X −→ U(X) onto the unit group of the enveloping quantale U(X). In × particular, this shows that the residuals of the unit group U(X) are the inverse residuals (13) of the quantale U(X). × Furthermore, X operates on X from the left and right via

− − ax = a 1 ; x; xa = a 1 → x; (23) such that the products ax and xa coincide with the corresponding products in U(X). The operations (23) satisfy

− ax → by = b(x → y)a 1 xa → ya = x → y − (24) xa ; yb = a 1(x ; y)b ax ; ay = x ; y

× for x; y ∈ X and a; b ∈ X .

Example 3.8. A unital quantum B-algebra X is a quantum BL-algebra [36] if x → u and x ; u are invertible for all x ∈ X. By [36, Theorem 4], quantum BL-algebras X can be characterized as twisted semidirect products G nδ Y of partially ordered × groups G with pseudo-BCK algebras Y such that G = X and Y = I(X).

1891 Quantum B-algebras

4. The group of connected components

X x; y ∈ X connected x z ; : : : ; z Let be a quantum B-algebra. We say that two elements are if there is a sequence = 0 n = y X z z z z z z i ∈ { ; : : : ; n} in with i−1 and i comparable (i.e. i−1 6 i or i−1 > i) for 1 . This deﬁnes an equivalence relation ∼ on X. The equivalence classes will be called the connected components of X. If x 6 y, then (6) implies that x 6 y 6 (y → z) ; z for all z ∈ X. Hence (1) gives

x 6 y =⇒ y → z 6 x → z: (25)

By symmetry, we also have x 6 y =⇒ y ; z 6 x ; z: (26)

Thus (4), (5), (25), and (26) show that ∼ is a congruence relation on X. So the connected components form a quantum B-algebra C(X) with trivial partial order, and there is a canonical morphism

c : X C(X): (27)

Deﬁnition 4.1. We call a quantum B-algebra X discrete if (27) is an isomorphism, i.e. if the partial order of X is trivial. We call X connected if |C(X)| = 1.

Of course, every group (with trivial partial order) is a discrete quantum B-algebra. Conversely,

Theorem 4.2. Every discrete quantum B-algebra is a group.

As by (6), a discrete quantum B-algebra satisﬁes (19), the theorem follows by Proposition 3.6. − In particular, the connected components of any quantum B-algebra form a group. The subalgebra Xu = c 1(u) will be called the unit component of X. We analyse the case where X can be reconstructed from C(X) and the unit component Xu. Let G be a group and X a quantum B-algebra. For a group homomorphism γ : G → Aut(X), we deﬁne the semidirect product G n X as the set of formal products ax with a ∈ G and x ∈ X, together with the partial order

ax 6 by ⇐⇒ a = b; x 6 y;

and the operations − a−1 − a−1b ax → by = ba 1(x → y) ; ax ; by = a 1b x ; y ; (28)

a − where x abbreviates γ(a 1)(x). It is easily veriﬁed that G nX is a quantum B-algebra. If X is unital, the semidirect product G nX is a special case of the G-extension G nδ X deﬁned in [36]. The map δ : X → {0; u} is given by

( u; x > u; δ(x) = 0; otherwise:

Assume that X is unital. Then there are natural embeddings G,→ G nX ←-X given by a 7→ au and x 7→ ux, × respectively. Thus G becomes a subgroup of the unit group (G nX) , and X a subalgebra that contains the unit component of G nX.

1892 W. Rump

Proposition 4.3. For a quantum B-algebra X, the following are equivalent.

× (a) X is unital, and the morphism (27) has a unital section, i.e. a morphism s: C(X) → X with image in X and cs = 1. ∼ (b) There exists a group G and a connected unital quantum B-algebra Y such that X = G nY .

Proof. (a) ⇒ (b): By assumption, the image of s is a subgroup G of the unit group of X. Consider the map − − γ : G → Aut(Xu) with γ(a)(x) = a 1 ; (a → x). By (23), this can be written as γ(a)(x) = axa 1. Furthermore, equations (22) imply that γ(a)(Xu) ⊂ Xu. Thus γ is a group homomorphism. Consider the map f : X → G nXu with − f(x) = sc(x)(sc(x) ; x). It is easily checked that g: G nXu → X with g(ax) = a 1 ; x = ax is inverse to f. Hence f is bijective. The implication (5) shows that g is monotonic. Therefore, (24) and (28) imply that g is an isomorphism of quantum B-algebras. × (b) ⇒ (a): By (28), X is unital, and G is a subgroup of (G nY ) . Since Y is connected, two elements ax; by ∈ G nY belong to the same connected component if and only if a = b. Hence c(ax) 7→ a gives a unital section of c : G nY C(G nY ).

Dudek and Jun [7] define a pseudo-BCI algebra to be an algebra (X; →; ;; u) with a reflexive antisymmetric binary relation 6 such that (6), (9), (10), and the following are satisfied:

x 6 y ⇐⇒ x → y = u ⇐⇒ x ; y = u: (29)

Equivalently,

Proposition 4.4. Every pseudo-BCI algebra is a unital quantum B-algebra. Conversely, a unital quantum B-algebra is a pseudo-BCI algebra if and only if its unit element u is maximal.

Proof. Let X be a pseudo-BCI algebra. Then (6) and (9) give

u 6 (u ; x) → x 6 (x → x) ; ((u ; x) → x) = u ; ((u ; x) → x) = u for all x ∈ X. Hence (u ; x) → x = u, and thus u ; x 6 x. On the other hand, (6) implies that x 6 (x → x) ; x = u ; x, which yields u ; x = x. By symmetry, u → x = u ; x = x. In particular, this shows that u is maximal. Hence

x 6 y ⇐⇒ u 6 x → y:

Therefore, (9) implies that 6 is a partial order. Assume that x 6 y ; z. Then (6) and (9) give y 6 (y ; z) → z 6 x → z. By symmetry, this yields (1). Furthermore, (9) gives x → (y ; z) 6 ((y ; z) → z) ; (x → z):

By (6) and (1), this implies that y 6 (y ; z) → z 6 (x → (y ; z)) → (x → z). Thus (1) gives x → (y ; z) 6 y ; (x → z). By symmetry, this yields (3). Finally, assume that y 6 z. Then x → y 6 (y → z) ; (x → z) = u ; (x → z) = x → z. This proves that X is a unital quantum B-algebra with u maximal. The converse is trivial.

Deﬁnition 4.5. We call a quantum B-algebra X normal if X is unital with x → x = x ; x = u for all x ∈ X.

The terminology is taken from algebraic number theory.

1893 Quantum B-algebras

Example 4.6. Let R be a noetherian integral domain with quotient ﬁeld K. An R-submodule A of K is said to be a fractional ideal of R if A is ﬁnitely generated with KA = K. The fractional ideals form a commutative semigroup A(R) with unit element R. Furthermore, A(R) is a residuated lattice with

A → B = (B : A) = {c ∈ K : cA ⊂ B};

hence a commutative unital quantum B-algebra. The ring R is normal, i.e. integrally closed, if and only if A(R) is normal in the sense of Deﬁnition 4.5.

Theorem 4.7. Let X be a normal quantum B-algebra. Then x → u = x ; u holds for all x ∈ X, and the map x 7→ x = x → u is a morphism from (X; →; ;; 6) to (X; ;; →; >). The image X is a partially ordered group.

Proof. For x ∈ X, we have

(x → u) → u = (x → u) → (x → u) ; (x → u) = (x → u) ; (x → u) → (x → u) = (x → u) ; u:

Hence (6) gives x 6 (x → u) → u, and thus x → u 6 x ; u. By symmetry, x → u = x ; u. For x; y ∈ X,

(x → y) → x = (x → y) → (x → (y ; y)) = (x → y) → (y ; (x → y)) = y ; (x → y) → (x → y) = y:

Therefore, x → y = (x → y) → (x ; x) = x ; ((x → y) → x) = x ; y:

By symmetry, this shows that x 7→ x is a morphism (X; →; ;; 6) → (X; ;; →; >). By [36, eq. (21)] we have x = x for all x ∈ X. For x; y ∈ X, this gives

(x → y) ; y = (x → y) ; (y → u) = y → x → y = y → (x ; y) = x ; (y → y) = x ; u = x:

Similarly, (x ; y) → y = x. By Proposition 3.6, this implies that X is a partially ordered group.

− For x ∈ X, we have x = x 1. Therefore, the image of x 7→ x is a subgroup X of X, hence a retract.

Corollary 4.8. Let X be a pseudo-BCI algebra. The subgroup X consists of the maximal elements of X. The map (27) induces an ∼ isomorphism X −→ C(X).

Proof. By (6), every maximal element x ∈ X satisﬁes x = x. Conversely, assume that x 6 y. Then x → y = u. Hence u = x → y = x ; y, which gives x 6 x 6 y. Thus y 6 y 6 x, which proves that x is maximal. The second assertion follows immediately.

× As every element a ∈ X satisﬁes a = a, we obtain the following corollary.

Corollary 4.9. The unit group X × of a pseudo-BCI algebra X is a subgroup of X, hence discrete.

The question arises whether the unit group always coincides with the subgroup X. Example 2.1 shows that this need not be the case.

1894 W. Rump

Theorem 4.10. Let X be a pseudo-BCI algebra. The following are equivalent. × (a) The subgroup X coincides with the unit group X . (b) X is a quantum BL-algebra. ∼ (c) There exists a group G and a pseudo-BCK algebra Y such that X = G nY .

The equivalence (a) ⇔ (b) follows by the deﬁnition of a quantum BL-algebra (Example 3.8). Proposition 4.3 gives the equivalence (a) ⇔ (c).

5. Quantum-B logic

The logic of pseudo-BCK algebras has been studied in Kühr’s habilitation thesis [26]. As usual, the unit element u stands for the “true” proposition, and the equivalences (29) enable us to transform an implication x 6 y between propositions x and y into x → y = u, which states that the proposition x → y is true. For a unital quantum B-algebra X, the unit element u need not be the greatest element, but (17) still guarantees the passage from x 6 y to the truth of x → y which is now to be written as u 6 x → y. In other words, the true propositions no longer have a ﬁxed value u, they are derivable from u, and u itself has become the smallest true proposition. If X is not unital, it suﬃces to assume local unitality.

Definition 5.1. We define a local unit of a quantum B-algebra X to be an element e ∈ X which satisfies e → e = e ; e = e. We say that X has enough local units if (x → x) ; x = (x ; x) → x = x (30) holds for all x ∈ X. We call X locally unital if X has enough local units and satisfies for all x; y ∈ X,

x → x 6 y =⇒ y → y 6 y: (31)

Note that by (30), for all e ∈ X, e → e = e ⇐⇒ e ; e = e: (32)

Proposition 5.2. Let X be a quantum B-algebra with enough local units. Then x → x and x ; x are local units for all x ∈ X.

Proof. By (31), we have (x → x) ; (x → x) = x → ((x → x) ; x) = x → x. Hence (32) implies that x → x is a local unit. The rest follows by symmetry.

Note that every unital quantum B-algebra is locally unital. In fact, x 6 u ; x implies u 6 x → x, which gives x 6 (x → x) ; x 6 u ; x = x. By symmetry, we get (30). Furthermore, x → x 6 y gives u 6 y = (y → y) ; y, and thus y → y 6 u → y = y. For a quantum B-algebra X with enough local units, x → x 6 x implies that x ; x 6 (x → x) ; x = x. Hence

x → x 6 x ⇐⇒ x ; x 6 x:

Therefore, we call an element x ∈ X positive if x → x 6 x and denote the set of positive elements by X +. Thus (31) states that X + is an upper set: If x 6 y and x ∈ X +, then y ∈ X +.

1895 Quantum B-algebras

Proposition 5.3. Let X be a quantum B-algebra with enough local units. Then, for x; y ∈ X,

x 6 y ⇐⇒ x → y ∈ X + ⇐⇒ x ; y ∈ X +: (33)

Proof. By symmetry, it suﬃces to prove the ﬁrst equivalence. Assume that x 6 y. Then (25) and (26) give y → y 6 x → y and (x → y) ; y 6 (y → y) ; y = y. Hence (x → y) ; (x → y) = x → ((x → y) ; y) 6 x → y, and thus x → y ∈ X +. Conversely, assume that x → y ∈ X +. Then (4) gives

y → y 6 (x → y) → (x → y) 6 x → y:

Hence (6) and (26) yield x 6 (x → y) ; y 6 (y → y) ; y = y.

Example 5.4. On the linearly ordered set X = {a; b; c; d} with a < b < c < d, consider the binary operation given by the table

→ a b c d a d d d d b a b d d c a a d d d a a c d

It is easily checked that X is a commutative quantum B-algebra with enough local units, but X is not unital. Furthermore, X + = {b; d}, which shows that X + need not be an upper set.

Example 5.5. Let X = {0; a; b; 1} be the nonlinear partially ordered set with smallest element 0 and greatest element 1. Together with the following binary operation, X becomes a commutative quantum B-algebra.

→ 0 a b 1 0 1 1 1 1 a 0 a 0 1 b 0 0 b 1 1 0 0 0 1

Here X + = {a; b; 1}, and thus X is locally unital, but not unital.

Logically, X + indicates the set of true propositions. Condition (31) states that X + is invariant under implication, i.e. the modus ponens x; x → y ` y (34)

holds for X +. The second equivalence of (33) gives the inference rules

x → y ` x ; y; x ; y ` x → y: (35)

1896 W. Rump

Now the quantum-B logic (qB) is given by the axioms

(B) (y → z) ; (x → y) → (x → z) ; (C) x → (y ; z) → y ; (x → z) ; 0 (C ) x ; (y → z) ; y → (x ; z) ; (I) x → x; together with the inference rules (34) and (35). The set L of propositions in (qB) is built recursively from a set V of countably many variables x; y; z; : : : by means of the operations → and ;, i.e. (1) each variable belongs to L; (2) if α; β ∈ L, then α → β ∈ L and α ; β ∈ L. A proposition α ∈ L is derivable from a subset Γ ⊂ L, written Γ ` α, if α can be obtained in finitely many steps as follows. (1) Every β ∈ Γ and every axiom is derivable; (2) if β is derivable, then every proposition obtained by substitution of the variables in β by arbitrary propositions is derivable; (3) if β and β → γ are derivable, then γ is derivable; (4) β → γ is derivable if and only if β ; γ is derivable. If Γ is empty, we write ` α instead of ∅ ` α. The set of α ∈ L with ` α will be denoted by L+. Let (X; →; ;; 6) be an algebra with two binary operations → and ; and a partial order 6 which satisfies (1) and (31). Thus → and ; determine each other. Every function v : V → X has a unique extension ev : L → X with ev (α → β) = v α → v β v α ; β v α ; v β ⊂ L X model α → β ∈ L+ {γ ∈ L ` γ} e( ) e( ) and e( ) = e( ) e( ). For Γ , we call a of (qB) + Γ if Γ = :Γ implies ev(α) 6 ev(β) for all α; β ∈ L and all maps v : V → X. If Γ = ∅, we call X a model of (qB). We define a preorder α β ⇐⇒ α → β ∈ L+ 6 Γ (36)

L L L+ on Γ = with Γ as positive cone. Axiom (B) implies that

β → γ 6 (α → β) → (α → γ) (37)

α; β; γ ∈ L L+ β α ∈ L+ ⇒ β ∈ L+ holds for Γ. Furthermore, the modus ponens (34) implies that Γ is an upper set: > Γ Γ . β → γ ∈ L+ α → β → α → γ ∈ L+ L So Γ gives ( ) ( ) Γ , which shows that Γ satisﬁes (4). Together with axiom (I), it follows L that (36) is a preorder on Γ. 0 L α β → γ β → α ; γ α ; β → γ ∈ L+ Now (C) and (C ) imply that Γ satisﬁes (3). Therefore, 6 implies that ( ) > ( ) Γ , β α ; γ L which yields 6 . By symmetry, it follows that (1) holds in Γ. So the inequality (37) can be written as

α → β 6 (β → γ) ; (α → γ):

Consequently, α 6 β implies β → γ 6 α → γ. Together with (4), and by symmetry, this implies that the equivalence α ∼ β α β α L →; ; L L /∼ relation Γ , deﬁned by 6 6 , is a congruence in ( Γ; ). Hence Γ = Γ is a quantum B-algebra. We L+ L+/∼ L L+ set Γ = Γ . If Γ = ∅, we write and , respectively.

Proposition 5.6. The algebras L are models of . Γ (qB) + Γ

L Proof. We have already shown that (36) is a partial order on Γ which satisﬁes (1). To verify (31), note that β → β → β ; β β → β ; β → β ∈ L+ β ∈ L α → α β β ∈ L+ (( ) ) = ( ) ( ) Γ holds for all Γ. Hence 6 implies that Γ , which β → β ; β ∈ L+ β → β β V → L x ∈ V yields ( ) Γ , that is, 6 . Now a map Γ is just given by substitution of variables by L L (equivalence classes of) propositions in Γ. Thus Γ is a model of (qB) + Γ.

In particular, Proposition 5.6 shows that conditions (1) and (31) are natural for the concept of a model of (qB). We call a partial order 6 of an algebra (X; →; ;) admissible if it satisﬁes (1) and (31). For such a partial order, the algebra X is a model of (qB) if and only if for every map v : V → X, the extension ev is monotonic, i.e. ev induces a morphism L → X of (→; ;; 6)-algebras.

1897 Quantum B-algebras

Proposition 5.7. Let (X; →; ;; 6) be an algebra with two binary operations → and ; and an admissible partial order 6. For a set Γ ⊂ L of additional axioms, X is a model of (qB) + Γ if and only if X is a locally unital quantum B-algebra with ev(Γ) ⊂ X + for all evaluations v : V → X.

Proof. Let X be a model of (qB) + Γ. As L is a quantum B-algebra, this implies that X satisﬁes (3) and (6). Since (y ; x) → (y ; x) ∈ L+, axiom (C) yields y ; ((y ; x) → x) ∈ L+. In particular,

(x → x) ; ((x → x) ; x) → x ∈ L+:

Thus (I), (34), and (35) yield ((x → x) ; x) → x ∈ L+. So we get (x → x) ; x 6 x 6 (x → x) ; x for all x ∈ X. By symmetry, this proves (30). To verify (4), assume that x; y; z ∈ X and y 6 z. Then y 6 z = (z → z) ; z. By (1), this implies that z → z 6 y → z. Furthermore, axiom (B) gives y → z 6 (x → y) → (x → z). Hence z → z 6 y → z 6 (x → y) → (x → z), and thus (1) and (3) yield x → y 6 (z → z) ; (x → z) = x → ((z → z) ; z) = x → z. This proves that X is a locally unital quantum B-algebra. α ∈ α → α → α ; α ∈ L+ α → α ; α ∈ L+ V → X If Γ, then (( ) ) implies that ( ) Γ . For any map , this gives ev(α) → ev(α) 6 ev(α), and thus ev(α) ∈ X +. Conversely, let X be a locally unital quantum B-algebra, and let v : V → X be any map. Assume that ev(Γ) ⊂ X +. We v L+ ⊂ X + α ∈ L+ α y → z ; x → y → x → z show that e( Γ ) . Thus let Γ be given. In case that = ( ) (( ) ( )), then (7) and (33) imply that ev(α) ∈ X +. For α = (x → (y ; z)) → (y ; (x → z)) or α = (x ; (y → z)) ; (y → (x ; z)), we get ev(α) ∈ X + by (3). Thus ev(α) ∈ X + if α is an axiom or α ∈ Γ. This does not change if any substitution of variables β; β → γ ∈ L+ v β ; v β → γ ∈ X + v β → v β v β v γ is applied. If Γ and e( ) e( ) has been shown, then e( ) e( ) 6 e( ) 6 e( ). Hence (31) v γ → v γ v γ v γ ∈ X + v L+ ⊂ X + α → β ∈ L+ implies that e( ) e( ) 6 e( ). Thus e( ) . This proves that e( Γ ) . Therefore, if Γ , then ev(α) → ev(β) ∈ X +, and (33) yields ev(α) 6 ev(β). Hence X is a model of (qB) + Γ.

As a particular case, we obtain

Corollary 5.8. Let (X; →; ;; 6) be an algebra with two binary operations → and ; and an admissible partial order 6. Then X is a model of (qB) if and only if X is a locally unital quantum B-algebra.

Furthermore, we have the following completeness theorem. Assume that Γ ⊂ L and α ∈ L. We write Γ α if ev(α) ∈ X + for every model X of (qB) + Γ and every map v : V → X.

Corollary 5.9. Let Γ ⊂ L and α ∈ L be given. Then Γ ` α iﬀ Γ α.

⇒ α L Proof. The implication “ ” has just been proved. Conversely, assume that Γ . By Proposition 5.6, Γ is a model of (qB) + Γ. Hence Γ ` α.

References

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