A few notes on quantum \(B\)-algebras
\par There exists the concept of quantum \(B\)-algebra of, \textit{W. Rump} and \textit{Y. C. Yang} [Ann. Pure Appl. Logic 165, No. 2, 759--785 (2014; Zbl 1322.03049)], which provides a unified semantics for non-commutative algebraic logic. A \textit{quantum \(B\)-algebra} is a partially ordered set (poset) \(X\) equipped with two binary operations \(\rightarrow\) and \(\rightsquigarrow\), which satisfy the following three conditions for every \(x,y,z\in X\): (1)~if \(y\leqslant z\), then \(x\rightarrow y\leqslant x\rightsquigarrow z\); (2) \(x\leqslant y\rightarrow z\) if and only if \(y\leqslant x\rightsquigarrow z\); and (3) \(x\rightsquigarrow(y\rightarrow z) = y\rightarrow(x\rightsquigarrow z)\). The present paper introduces the notion of generalized quantum \(B\)-algebra and considers its properties, some of which are related to the respective properties of quantum \(B\)-algebras. \par A \textit{generalized quantum \(B\)-algebra} is a poset \(X\) equipped with two binary operations \(\rightarrow\) and \(\rightsquigarrow\), which satisfy conditions (1) and (2) mentioned above. A \textit{morphism} of generalized quantum \(B\)-algebras \(f:X\rightarrow Y\) is an order-preserving map, which satisfies the following two equivalent conditions for every \(x_1,x_2\in X\): \(f(x_1\rightarrow x_2)\leqslant f(x_1)\rightarrow f(x_2)\) and \(f(x_1\rightsquigarrow x_2)\leqslant f(x_1)\rightsquigarrow f(x_2)\). The present paper shows that the category \textbf{GqBAlg} of generalized quantum \(B\)-algebras and their morphisms is dually equivalent to the category \textbf{LogPQ} of logical pre-quantales, where a \textit{pre-quantale} is a \(\bigvee\)-semilattice \(Q\) equipped with a binary operation \(\cdot\), which distributes over arbitrary \(\bigvee\) from both sides, i.e., \(u\cdot(\bigvee S)=\bigvee_{s\in S}u\cdot s\) and \((\bigvee S)\cdot u=\bigvee_{s\in S}s\cdot u\) for every \(u\in Q\) and every \(S\subseteq Q\) [\textit{K. I. Rosenthal}, Quantales and their applications. Harlow: Longman Scientific \&| Technical; New York: John Wiley \&| Sons, Inc. (1990; Zbl 0703.06007)]. \par A generalized quantum \(B\)-algebra \(X\) is said to be \textit{unital} provided that there exists an element \(u\in X\) such that \(u\rightarrow x=u\rightsquigarrow x\) for every \(x\in X\). The authors provide a characterization of unital generalized quantum \(B\)-algebras in terms of unital residuated semigroups of, e.g., [\textit{G. Birkhoff}, Lattice theory. Corr. repr. of the 1967 3rd ed. Providence, RI: American Mathematical Society (AMS) (1979; Zbl 0505.06001)]. \par The authors also introduce the concept of \textit{\(C\)-algebra} as a set \(X\) equipped with two binary operations \(\rightarrow\) and \(\rightsquigarrow\) satisfying condition~(3) mentioned above. An element \(d\) of a \(C\)-algebra \(X\) is said to be \textit{dualizing} provided that \((x\rightarrow d)\rightsquigarrow d = x = (x\rightsquigarrow d)\rightarrow d\) for every \(x\in X\). The authors show that a \(C\)-algebra \(X\) is a group if and only if \(X\) is non-empty, and every element of \(X\) is dualizing. \par The authors finally consider some properties of quantum \(B\)-algebras induced by the so-called \textit{zig-zag relation} of, e.g., [\textit{T. S. Blyth}, Lattices and ordered algebraic structures. London: Springer (2005; Zbl 1073.06001)]. \par The paper is well written, provides nearly all of its required preliminaries (the omitted concepts can be easily found in the suggested references), and will be of interest to all the researchers studying algebraic structures related to logic.
scientific article; zbMATH DE number 7450677
scientific article