Non-commutative logical algebras and algebraic quantales

Rump, Wolfgang; Yang, Yi

doi:10.1016/j.apal.2013.10.007

Cited by 40 publications

(52 citation statements)

References 28 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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).…”

Section: Definition 12mentioning

confidence: 99%

“…In this section, we briefly review the logical aspects of quantales which led to the concept of quantum B-algebra [36]. …”

Section: Non-commutative Logic and Quantalesmentioning

confidence: 99%

“…Now assume that X is unital, that is, X contains an element which makes U(X ) into a unital quantale. By [36,Theorem 3], the whole unit group U(X )…”

Section: 23 224])mentioning

confidence: 99%

“…The quantum B-algebra X can then be identified with the set of logical [36] quantale morphisms Q → Q(1) onto the endomorphism quantale Q(1) of the poset {0 1} with 0 < 1. The multiplication of the quantale Q corresponds to a ternary relation in X .…”

Section: 23 224])mentioning

confidence: 99%

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

Section: Definition 12mentioning

confidence: 99%

See 4 more Smart Citations

Quantum B-algebras

Rump

2013

Open Mathematics

Self Cite

View full text Add to dashboard Cite

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 manifests 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

show abstract

“…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).…”

Section: Definition 12mentioning

confidence: 99%

“…In this section, we briefly review the logical aspects of quantales which led to the concept of quantum B-algebra [36]. …”

Section: Non-commutative Logic and Quantalesmentioning

confidence: 99%

“…Now assume that X is unital, that is, X contains an element which makes U(X ) into a unital quantale. By [36,Theorem 3], the whole unit group U(X )…”

Section: 23 224])mentioning

confidence: 99%

Section: 23 224])mentioning

confidence: 99%

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

Section: Definition 12mentioning

confidence: 99%

See 3 more Smart Citations

Quantum B-algebras

Rump

2013

Open Mathematics

Self Cite

View full text Add to dashboard Cite

show abstract

Quantum B‐modules

Xia

Rump

2022

Mathematical Logic Qtrly

Self Cite

View full text Add to dashboard Cite

Quantum B-algebras are partially ordered algebras characterizing the residuated structure of a quantale. Examples arise in algebraic logic, non-commutative arithmetic, and quantum theory. A quantum B-algebra with trivial partial order is equivalent to a group. The paper introduces a corresponding analogue of quantale modules. It is proved that every quantum B-module admits an injective envelope which is a quantale module. The injective envelope is constructed explicitly as a completion, a multi-poset version of the completion of Dedekind and MacNeille.

show abstract