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Non-commutative logical algebras and algebraic quantales. (English) Zbl 1322.03049

Quantum B-algebras are the partially ordered implicational algebras arising as subreducts of quantales. The authors show that the opposite of the category of quantum B-algebras is equivalent to the category of logical quantales, in the way that every quantum B-algebra admits a natural embedding into a logical quantale, the enveloping quantale. They prove that the unit group of the enveloping quantale of a quantum B-algebra \(X\) is always contained in \(X\), which gives a functorial subgroup \(X^X\) of \(X\). The results of N. Galatos and C. Tsinakis [J. Algebra 283, No. 1, 254–291 (2005; Zbl 1063.06008)] and B. Jónsson and C. Tsinakis [Stud. Log. 77, No. 2, 267–292 (2004; Zbl 1072.06003)] on the splitting of generalized BL-algebras into a semidirect product of a partially ordered group operating on an integral residuated poset are extended to a characterization of twisted semidirect products of a po-group by a quantum B-algebra.

MSC:

03G27 Abstract algebraic logic
06F07 Quantales
20F60 Ordered groups (group-theoretic aspects)
06B23 Complete lattices, completions
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