Quantum B-algebras

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 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

Q-Filters of Quantum B-Algebras and Basic Implication Algebras

“…Definition 1. Let (X, ≤) be partially ordered set endowed with two binary operations → and [24,25]. Then, (X, →, , ≤) is called a quantum B-algebra if it satisfies: ∀x, y, z∈X,…”

“…Proposition 6. Every pseudo-BCI algebra is a unital quantum B-algebra [25]. And, a quantum B-algebra is a pseudo-BCI algebra if and only if its unit element u is maximal.…”

“…For formalizing the implication fragment of the logic of quantales, Rump and Yang proposed the notion of quantum B-algebras [24,25], which provide a unified semantic for a wide class of nonclassical logics. Specifically, quantum B-algebras encompass many implication algebras, like pseudo-BCK/BCI algebras, (commutative and noncommutative) residuated lattices, pseudo-MV/BL/MTL algebras, and generalized pseudo-effect algebras.…”

Filters in Strong BI-Algebras and Residuated Pseudo-SBI-Algebras

“…For example, algebraic structures with one implication operator, BCK/BCI algebras, BI-algebras, etc. ; algebraic structures with double implication operators, quantum B-algebras, pseudo-BCK/BCI algebras [6][7][8][9], etc. ; algebraic structures with binary operators and implication operators, residuated lattices, non-associative residuated lattices, etc.…”

Filters on Some Classes of Quantum B-Algebras

“…By definition, a quantale is a complete lattice Q with an associative multiplication · that distributes over arbitrary joins. By the completeness of Q, there are left and right adjoint operations (called residuals) → and of · such that Note that, as was mentioned in [26] and [27], in any quantale the following conditions are satisfied:…”