A few notes on quantum \(B\)-algebras
type: Item
instance of: scholarly article
title: A few notes on quantum \(B\)-algebras
zbMATH Open document ID: 1482.06002
author: Xiaoting Xu
author: Shengwei Han
published in: Studia Logica
publication date: 2021-12-27T00:00:00Z
review text: \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 applicatio…
Mathematics Subject Classification ID: 06F07
Mathematics Subject Classification ID: 03G27
Mathematics Subject Classification ID: 06A11
Mathematics Subject Classification ID: 06F05
Mathematics Subject Classification ID: 18F75
Mathematics Subject Classification ID: 20N02
zbMATH DE Number: 7450677
zbMATH Keywords: algebraic logic
zbMATH Keywords: balanced element
zbMATH Keywords: \(C\)-algebra
zbMATH Keywords: dualizing element
zbMATH Keywords: embedding
zbMATH Keywords: (generalized) quantum \(B\)-algebra
zbMATH Keywords: injective object
zbMATH Keywords: logical quantale
zbMATH Keywords: positive subset
zbMATH Keywords: pre-quantale
zbMATH Keywords: quantale
zbMATH Keywords: residuated semigroup
zbMATH Keywords: submultiplicative map
zbMATH Keywords: supercompact element
zbMATH Keywords: zig-zag relation
reviewed by: Sergejs Solovjovs
MaRDI profile type: Publication
full work available at URL: s11225-021-09953-2
OpenAlex ID: W3171389437
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DOI: 10.1007/S11225-021-09953-2
description: scientific article; zbMATH DE number 7450677