# Quantum B-algebras: their omnipresence in algebraic logic and beyond

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## Abstract

Quantum B-algebras are implicational subreducts of quantales. Their ubiquity and unifying rôle in algebraic logic and beyond is discussed in a brief survey, with old and new examples included. A special section is devoted to *CKL*-algebras (alias HBCK-algebras). Using a natural embedding of any *CKL*-algebra into a semibrace, Cornish’s identity is derived, which yields a new syntactic proof of Wroński’s conjecture that *CKL*-algebras form a variety.

### Keywords

Quantum B-algebra CKL-algebra Semibrace Residuated poset Partially ordered group## 1 Introduction

Quantum B-algebras were introduced in Rump (2013) and studied further in Rump and Yang (2014). Their ubiquity and unifying nature in the realm of algebraic logic was established by three main classes of algebras. Prototypes for these classes are BCK-algebras, residuated posets, and effect algebras. On the other hand, it was proved that quantum B-algebras formalize the implicational part of quantales. Recently, the *completion*\(\widehat{X}\) of a quantum B-algebra *X* was constructed (Rump 2016), a quantale that can be viewed as an injective envelope of *X*. Some concepts for quantales survive in the framework of quantum B-algebras. Although the latter need not be multiplicatively closed, it makes sense to speak of a *unital* quantum B-algebra, with a *unit element* related to that of the ambient quantale.

The present paper starts with a brief survey on basic features of not necessarily unital quantum B-algebras, with old and new examples. In a third section, we focus upon a class of quantum B-algebras occurring in connection with Wroński’s conjecture on naturally ordered BCK-algebras.

Roughly speaking, a quantum B-algebra is a partially ordered set with two binary operations \(\rightarrow \) and \(\leadsto \) satisfying three conditions (see Sect. 2). Logically, \(\rightarrow \) and \(\leadsto \) represent the left and right implications, while \(\leqslant \) stands for the entailment relation. In commutative logic, the two implications \(\rightarrow \) and \(\leadsto \) coincide. As already shown in Rump (2013), a quantum B-algebra with trivial partial order is equivalent to a group. Then \(u:=x\rightarrow x\) does not depend on *x* and gives the unit element of the group. In particular, such a quantum B-algebra is always unital. We also consider the complementary case where the quantum B-algebra reduces to a mere poset. In this way, any partially ordered set with a greatest element *u* is a unital quantum B-algebra with unit element *u*.

Apart from the above-mentioned three prototypical specializations and building blocks of a unital quantum B-algebra, the *connected components* (Rump 2013) of every (not necessarily unital) quantum B-algebra *X* form a group *C*(*X*), which implies, in particular, that *X* has a distinguished component, according to the unit element of *C*(*X*). This can be applied to determine the structure of a pseudo-BCI algebra (see Rump 2013, Sect. 4). Here we exhibit another building block of *X*, namely, the set *G*(*X*) of *group-like* elements \(x\in X\), which satisfy \(x\rightarrow x=x\leadsto x\) and the above equations \((x\rightarrow y) \leadsto y=(x\leadsto y)\rightarrow y=x\) for all \(y\in X\). We show that *G*(*X*) is either empty or a partially ordered group (Theorem 1).

*L*

*-algebras*(Rump 2008a), which are named by the relation

*L*

*-algebra*

*X*has a

*logical unit*, that is, an element 1 which satisfies

*L*-algebra is a quantum B-algebra if and only if it is a BCK-algebra. Such algebras are known as HBCK-algebras (Blok and Ferreirim 1993) or

*CKL*

*-algebras*(Rump 2009). Traczyk (1988) studied

*CKL*-algebras as a special class of BCK-algebras. Wroński (1985) conjectured that

*CKL*-algebras coincide with the implicational subreducts of hoops. In particular, this means that Cornish’s identity (Cornish 1982)

*CKL*-algebra. Ferreirim (1992), Blok and Ferreirim (1993), Ferreirim (2001) confirmed Wroński’s conjecture by a careful analysis of subdirectly irreducible hoops and by adapting these results to BCK-algebras and

*CKL*-algebras. In this way, they obtained an indirect proof of Cornish’s identity. However, they have not been able to transform their findings into a syntactic proof. This problem was settled by Kowalski (1994), and independently, by Wos and Veroff (1994), using automated reasoning with Otter (McCune 1994). In Theorem 2, we provide a simple, syntactic, self-contained proof of (J) for arbitrary

*CKL*-algebras.

## 2 The category of quantum B-algebras

*quantum B-algebra*(Rump 2013; Rump and Yang 2014) is defined to be a partially ordered set

*X*with two binary operations \(\rightarrow \) and \(\leadsto \) satisfying

### Example 1

Let *G* be a quasigroup. Thus, for \(x,y\in G\), there are unique elements \(x\rightarrow y\) and \(x\leadsto y\) in *G* with \((x\rightarrow y)x=y= x(x\leadsto y)\). With the trivial partial order, we have \(x=y\rightarrow z\Longleftrightarrow xy=z \Longleftrightarrow y=x\leadsto z\). This proves (2), while (3) is trivial. To check what Eq. (1) is about, we set \(u:=y\leadsto z\) and \(v:=x\rightarrow z\). Then \(yu=z=vx\), and Eq. (1) is equivalent to \(x\rightarrow u=y\leadsto v\). With \(t:=x\rightarrow u=y\leadsto v\), this gives \(y(tx)=yu=vx=(yt)x\). Thus Eq. (1) implies that *G* is associative, whence it cannot be dropped, in general.

To see that (2) is not redundant, too, consider a set *X* with trivial partial order and \(x\rightarrow y=x\leadsto y=y\) for all \(x,y\in X\). Then (1) and (3) hold, but (2) fails if \(|X|>1\).

*Q*is said to be a

*quantale*(Mulvey 1986; Rosenthal 1990) if

*Q*is a complete lattice satisfying

*Q*such that the operations \(\rightarrow \) and \(\leadsto \) and the relation \(\leqslant \) are induced by the residuals

*Q*.

*residuated poset*, that is, a partially ordered semigroup with binary operations \(\rightarrow \) and \(\leadsto \) satisfying (9), is a quantum B-algebra.

*X*can be characterized as an element \(u\in X\) which satisfies

*X*, such an element

*u*is unique. If it exists,

*X*is called

*unital*. On the other hand, a residuated poset

*X*is commutative if and only if it satisfies

*commutative*.

## 3 Further examples

In classical algebra, most structures are based on semigroups. When partial orders arise, residuals are notably absent. In every first course on group theory, however, subgroups are characterized as non-empty subsets *H* for which \(a,b\in H\) implies that *H* also contains \(ab^{-1}\). Well, this is a residual!

### Example 2

*G*be a group. With

*G*is a quantum B-algebra with trivial partial order.

In Rump (2013), Theorem 4.2, we proved the converse:

### Proposition 1

A quantum B-algebra *X* is a group if and only if its partial order is trivial.

This was obtained as a consequence of a more general theorem. For convenience, we give a direct proof.

### Proof

*X*be a quantum B-algebra with trivial partial order. Assume that the equation \(z\rightarrow x=z\rightarrow y\) holds for given \(x,y,z\in X\). Then (5) implies that

*u*for this unique element \(x\rightarrow x= y\leadsto y\). Then \(x\rightarrow ux=u=x\rightarrow x\) yields \(ux=x\), and \(x=(x\leadsto x)\rightarrow x= u\rightarrow x\) implies that \(xu=x\) for all \(x\in X\). Finally, \(x\rightarrow u=x\rightarrow (x\rightarrow u)x\) gives \((x\rightarrow u)x=u\), which shows that every element of

*X*is invertible. \(\square \)

### Example 3

Let us discuss some generalizations of Examples 2 and 3.

### Example 4

Based on this example, we give the following

### Definition 1

Let *X* be a quantum B-algebra. We call an element \(x\in X\)*group-like* if it satisfies \(x\rightarrow x=x\leadsto x\) and Eq. (14) for all \(y\in X\).

### Theorem 1

Let *X* be a quantum B-algebra. The set *G*(*X*) of group-like elements is either empty or a partially ordered subgroup of *X*.

### Proof

*right admissible*if

*left admissible*if \((x\rightarrow y)\leadsto y=x\) holds for all \(y\in X\). Assume that \(x,y\in X\) are group-like. Then \(x\rightarrow y\) is right admissible. By the dual of the above argument, \(u:=x\rightarrow x=x\leadsto x\) is left admissible. Hence

*x*.

*G*(

*X*) is a quantum B-subalgebra of

*X*. Now Rump (2013), Proposition 3.6, implies that

*G*(

*X*) is a partially ordered group if \(G(X)\not =\varnothing \). \(\square \)

*X*be a set with a binary operation \(\rightarrow \). An element \(1\in X\) is said to be a

*logical unit*(Rump 2008a) if

*X*is said to be a

*pseudo-BCK-algebra*(Georgescu and Iorgulescu 2001; van Alten 2006; Dvurečenskij and Kühr 2009) if for \(x,y,z\in X\),

### Example 5

*X*satisfies \(x\rightarrow y=1\Longleftrightarrow x\leadsto y=1\). Therefore, we can introduce a partial order

*X*into a quantum B-algebra. Conversely, by Rump and Yang (2014), Corollary 2 of Proposition 12, a quantum B-algebra

*X*is pseudo-BCK if and only if

*X*has a greatest element which is a unit element. In particular, every partially ordered set with greatest element 1 is a pseudo-BCK-algebra unit element 1.

*X*is pseudo-BCK if and only if \(|X|=1\). There is a common generalization. Namely, an algebra \((X;\rightarrow ,\leadsto )\) with a partial order \(\leqslant \) and a constant

*u*is called a

*pseudo-BCI algebra*(Jun et al. 2006) if it satisfies

### Example 6

Every pseudo-BCI algebra is a unital quantum B-algebra. Conversely, by Rump (2013), Proposition 4.4, a unital quantum B-algebra is pseudo-BCI if and only if its unit element is maximal.

More generally, Theorem 1 yields

### Corollary 1

Let \(X\not =\varnothing \) be a commutative quantum B-algebra. Assume that every element of *X* is contained in a maximal element. Then the maximal elements form a subgroup of *X* which coincides with *G*(*X*).

### Proof

Every maximal element of *X* is group-like. Hence *G*(*X*) is non-empty and contains a maximal element *t*. Suppose that \(x,y\in G(X)\) satisfy \(x\leqslant y\). Then \(t\leqslant tx^{-1}y\), which yields \(t=tx^{-1}y\). Hence \(x=y\). So the partial order of *G*(*X*) is trivial. Therefore, *G*(*X*) must coincide with the set of maximal elements in *X*. \(\square \)

### Corollary 2

Let *X* be a quantum B-algebra with a smallest element 0. Then *X* has a greatest element 1, and \(G(X)= \{1\}\). In particular, \(1\rightarrow 1=1\leadsto 1=1\).

### Proof

For all \(x,y\in X\), we have \(0\leqslant x\rightarrow y\). Hence \(x\leqslant 0\leadsto y\), which shows that \(1:=0\leadsto y\) is a greatest element. Thus \(1\rightarrow 1=1\rightarrow (0 \leadsto y)=0\leadsto (1\rightarrow y)=1\), and similarly, \(1\leadsto 1=1\). So 1 is group-like, and the argument in the preceding proof shows that the partial order of *G*(*X*) is trivial. Whence \(G(X)=\{1\}\). \(\square \)

One may ask whether a quantum B-algebra with a greatest element is pseudo-BCI. The following example shows that this need not be the case.

### Example 7

Consider the finite quantum B-algebra \(X=\{0,x,y,z, t,1\}\) given by the following tables and Hasse diagram

### Example 8

*X*can be obtained as a quantum B-subalgebra of the quantale given by the power set of the semigroup

Here \(G(X)=\{1\}\), but 1 is not a unit element of *X*. Thus *X* is neither pseudo-BCK nor even pseudo-BCI.

### Example 9

*G*by a pseudo-BCK-algebra

*X*(see Rump 2013, Sect. 4). For a group homomorphism \(\gamma :G\rightarrow \text{ Aut }(X)\), the semidirect product \(G\ltimes X\) consists of the formal products

*ax*with \(a\in G\) and \(x\in X\), with partial order

*twisted*semidirect products \(G\ltimes _\delta X\) of partially ordered groups with pseudo-BCK-algebras form a class of quantum B-algebras. These so-called

*quantum BL-algebras*(Rump and Yang 2014) are unital quantum B-algebras which are characterized by the property that \(x\rightarrow u\) and \(x\leadsto u\) are

*invertible*for all

*x*in the sense of Rump and Yang (2014), Definition 8. For any quantum B-algebra

*X*, these invertible elements form a subgroup of

*X*, the

*unit group*\(X^\times \). In general, \(X^\times \) is a proper subgroup of

*G*(

*X*).

More examples and classes of quantum B-algebras can be found in Rump (2013), Rump and Yang (2014), Rump (2016). In the following section, we deal with a special class of commutative quantum B-algebras.

## 4 Quantum B-algebras versus L-algebras

*X*with a binary operation \(\rightarrow \) is said to be a

*cycloid*(Rump 2008a) if the equation

*X*has a logical unit 1 with

*X*is called an

*L*

*-algebra*(Rump 2008a). Every

*L*-algebra is partially ordered by

*L*-algebras are

*left hoops*, that is, monoids

*H*with an additional binary operation \(\rightarrow \) satisfying

*H*is commutative as a semigroup, then

*H*is said to be a

*hoop*(Büchi and Owens 1975; Blok and Raftery 1997; Blok and Ferreirim 1993).

There is an intermediate structure between *L*-algebras and left hoops.

### Definition 2

*semibrace*(Rump 2008b) if

A semibrace is equivalent to a *brace* (Rump 2007) if and only if \((A,\wedge )\) is a group. Braces are ring-like structures which are closely related to set-theoretic solutions of the quantum Yang–Baxter equation Etingof et al. (1999), Lu et al. (2000), Rump (2005). Every semibrace *X* is a cycloid, but not necessarily an *L*-algebra. If *X* is an *L*-algebra, we call *X* an *L**-semibrace*. Then 1 is a logical unit, and *X* is a \(\wedge \)-semilattice. By Rump (2008a), Proposition 4, every left hoop is an *L*-semibrace.

*L*-algebra satisfies the identity

*L*-algebra

*X*which satisfies

*CKL-algebra*(Rump 2009). Note that any

*CKL*-algebra also satisfies

*CKL*-algebra is a commutative unital quantum B-algebra with unit element 1.

*CKL*-algebras are equivalent to

*naturally ordered*BCK-algebras, that is, BCK-algebras satisfying (L). As an implicational version of the basic equation for a hoop, Eq. (L) has also been denoted by (H). Accordingly,

*CKL*-algebras have been called

*HBCK-algebras*(Blok and Ferreirim 1993). A conjecture of Wroński (1985) states that

*CKL*-algebras form a variety and coincide with the implicational subreducts of hoops. These conjectures were proved by Ferreirim (1992), Blok and Ferreirim (1993), Ferreirim (2001). In particular, they proved that the variety of

*CKL*-algebras is given by identities (16), (B), (C), (L) together with Cornish’s identity (Cornish 1982)

There are two natural ways to embed a *CKL*-algebra *X* into a larger structure. Firstly, as an *L*-algebra, *X* has a *self-similar closure* (Rump 2008a), that is, *X* can be represented as an *L*-subalgebra of a *self-similar* left hoop *S*(*X*) such that *X* generates *S*(*X*) as a monoid. By Rump (2008a), Theorem 3, the self-similar closure *S*(*X*) is unique, up to isomorphism. Furthermore, Rump (2008a), Proposition 10, implies that *S*(*X*) satisfies (K). However, (C) does not carry over to *S*(*X*), or equivalently, *S*(*X*) need not be commutative. Secondly, as a quantum B-algebra, *X* has a *completion* (Rump 2016), that is, *X* embeds *densely* into a quantale \(\widehat{X}\). In particular, this means that every element of \(\widehat{X}\) is of the form \(\bigwedge A\) for a subset \(A\subset X\). Hence \(\widehat{X}\) is commutative.

None of the embeddings of *X* into *S*(*X*) or \(\widehat{X}\) immediately yields an embedding into a hoop: *S*(*X*) is a left hoop, not necessarily commutative, while \(\widehat{X}\) is a commutative quantale, but not necessarily a hoop. Thus it is tempting to believe that a natural embedding into a hoop might still be found between the Scylla of non-commutative hoops and the Charybdis of commutative non-hoops. This would be a strong improvement since Blok and Ferreirim’s construction is highly non-canonical. Roughly speaking, they reduce the problem to finitely generated subdirectly irreducible *CKL*-algebras and show that every *n*-generated subdirectly irreducible *CKL*-algebra splits into an ordinal sum \(F\vee S\) where *F* is \((n-1)\)-generated and *S* is totally ordered. So *F* comes within the inductive hypothesis, while *S* is a subreduct of an MV-algebra.

As (J) holds in any hoop, the result of Blok and Ferreirim yields an indirect proof that (J) is valid for any *CKL*-algebra. Kowalski (1994) obtained a (rather intricate) syntactic proof. Independently, Veroff got an automated proof (see Wos and Veroff 1994, Sect. 7.3) using Otter (McCune 1994). In what follows, we give an alternative, hopefully more intelligible proof.

By Rump (2008b), Theorem 3, every *L*-algebra naturally embeds into an *L*-semibrace *C*(*L*) which is uniquely determined, up to isomorphism, by the property that every element of *C*(*X*) is a finite meet \(x_1\wedge \cdots \wedge x_n\) with \(x_i\in X\). By Rump (2008b), Corollary 2 of Theorem 3, *C*(*X*) coincides with the \(\wedge \)-subsemilattice of *S*(*X*) generated by *X*. Note, however, that existing meets in *X* need not be respected by the embedding \(X\hookrightarrow C(X)\).

### Proposition 2

Let *X* be a *CKL*-algebra. Then *C*(*X*) is a *CKL*-algebra.

### Proof

*S*(

*X*) satisfies (K). Hence (K) holds in

*C*(

*X*). So it remains to verify Eq. (C) in

*C*(

*X*). By (18), we can assume that \(z\in X\). So we have to verify

### Proposition 3

Let *X* be a *CKL*-algebra and \(x,y,z\in X\) with \(z\leqslant y\). For \(u\in X\), we set \(u':=u\rightarrow z\). Then \(x\rightarrow y''=x''\rightarrow y''=(x\rightarrow y)''\).

### Proof

Now we are ready to prove

### Theorem 2

Every *CKL*-algebra satisfies (J).

### Proof

*C*(

*X*) is a

*CKL*-algebra. We set \(z:=x\wedge y\in C(X)\). With the abbreviation of Proposition 3, this gives

*S*(

*X*). Thus (K) yields

*S*(

*X*) we have

*x*and

*y*. \(\square \)

## Notes

### Compliance with ethical standards

### Conflict of interest

The authors confirm that there is no conflict of interest.

### References

- Blok WJ, Ferreirim IMA (1993) Hoops and their implicational reducts (abstract). In: Algebraic methods in logic and in computer science (Warsaw, 1991). Banach Center Publication, 28, Polish Academy Science, Warsaw, pp 219–230Google Scholar
- Blok WJ, Raftery JG (1997) Varieties of commutative residuated integral pomonoids and their residuation subreducts. J Algebra 190:280–328MathSciNetCrossRefMATHGoogle Scholar
- Bosbach B (1982) Residuation groupoids. Results Math 5:107–122MathSciNetCrossRefMATHGoogle Scholar
- Büchi JR, Owens TM (1975) Complemented monoids and hoops. Manuscript, unpublishedGoogle Scholar
- Cignoli R, Torrens Torrell A (2004) Glivenko like theorems in natural expansions of BCK-logic. Math Log Q 50:111–125MathSciNetCrossRefMATHGoogle Scholar
- Cornish WH (1982) BCK-algebras with a supremum. Math Japon 27(1):63–73MathSciNetMATHGoogle Scholar
- Dvurečenskij A, Kühr J (2009) On the structure of linearly ordered pseudo-BCK-algebras. Arch Math Log 48(8):771–791MathSciNetCrossRefMATHGoogle Scholar
- Etingof P, Schedler T, Soloviev A (1999) Set-theoretical solutions to the quantum Yang–Baxter equation. Duke Math J 100:169–209MathSciNetCrossRefMATHGoogle Scholar
- Ferreirim IMA (1992) On varieties and quasivarieties of hoops and their reducts. Ph.D. Thesis, University of Illinois at ChicagoGoogle Scholar
- Ferreirim IMA (2001) On a conjecture by Andrzej Wroński for BCK-algebras and subreducts of hoops. Sci Math Jpn 53(1):119–132MathSciNetCrossRefMATHGoogle Scholar
- Georgescu G, Iorgulescu A (2001) Pseudo-BCK algebras: an extension of BCK algebras. In: Combinatorics, computability and logic (Constanţa, 2001). Springer, London, pp 97–114 Springer Series in Discrete Mathematics and Theoretical Computer ScienceGoogle Scholar
- Jun YB, Kim HS, Neggers J (2006) On pseudo-BCI ideals of pseudo-BCI algebras. Mat Vesnik 58(1–2):39–46MathSciNetMATHGoogle Scholar
- Kowalski T (1994) A syntactic proof of a conjecture of Andrzej Wroński. Rep Math Log 28:81–86MATHGoogle Scholar
- Lu J-H, Yan M, Zhu Y-C (2000) On the set-theoretical Yang–Baxter equation. Duke Math J 104:1–18MathSciNetCrossRefMATHGoogle Scholar
- McCune W (1994) Otter 3.0, reference manual and guide. Technical report ANL-94/6, Argonne National Laboratory, Argonne, IllinoisGoogle Scholar
- Mulvey CJ (1986) & Second topology conference (Taormina, 1984). Rend Circ Mat Palermo (2) 12:99–104Google Scholar
- Rosenthal KI (1990) Quantales and their applications, Pitman research notes in mathematics series 234. Longman Scientific & Technical, Harlow (copublished in the United States with John Wiley & Sons Inc., New York)Google Scholar
- Rump W (2005) A decomposition theorem for square-free unitary solutions of the quantum Yang–Baxter equation. Adv Math 193:40–55MathSciNetCrossRefMATHGoogle Scholar
- Rump W (2007) Braces, radical rings, and the quantum Yang–Baxter equation. J Algebra 307:153–170MathSciNetCrossRefMATHGoogle Scholar
- Rump W (2008a) \(L\)-algebras, self-similarity, and \(l\)-groups. J Algebra 320(6):2328–2348Google Scholar
- Rump W (2008b) Semidirect products in algebraic logic and solutions of the quantum Yang–Baxter equation. J Algebra Appl 7(4):471–490Google Scholar
- Rump W (2009) A general Glivenko theorem. Algebra Univers 61(3–4):455–473MathSciNetCrossRefMATHGoogle Scholar
- Rump W (2013) Quantum B-algebras. Cent Eur J Math 11(11):1881–1899MathSciNetMATHGoogle Scholar
- Rump W (2016) The completion of a quantum B-algebra. Cah Topol Géom Différ Catég 57(3):203–228MathSciNetMATHGoogle Scholar
- Rump W, Yang Y (2014) Non-commutative logical algebras and algebraic quantales. Ann Pure Appl Log 165(2):759–785MathSciNetCrossRefMATHGoogle Scholar
- Traczyk T (1988) On the structure of BCK-algebras with \(zx\cdot yx=zy \cdot xy\). Math Japon 33(2):319–324MathSciNetMATHGoogle Scholar
- van Alten CJ (2006) On varieties of biresiduation algebras. Stud Log 83(1–3):425–445MathSciNetCrossRefMATHGoogle Scholar
- Wos L, Veroff R (1994) Logical basis for the automation of reasoning: case studies, vol 2. Oxford Science Publication, Oxford University Press, New York, pp 1–40. Handbook of logic in artificial intelligence and logic programmingGoogle Scholar
- Wroński A (1985) An algebraic motivation for BCK-algebras. Math Japon 30(2):187–193MathSciNetMATHGoogle Scholar