Quantum B‐modules

Abstract

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.

A categorical equivalence between logical quantale modules and quantum B‐modules

“…">2.Let $$(_{X}A,\twoheadrightarrow , \rightarrowtail )$$ be a quantum B‐module, and let $$U(X)$$ and $$U(A)$$ be upper sets of X and A , respectively. It was shown in [23] that $$_{U(X)}U(A)$$ is a quantale module with respect to set‐theoretic union and the multiplications $$$$\begin{align*} X_1 \cdot X_2&=\lbrace x \in X \mid \exists x_2 \in X_2 : x_2 \rightarrow x \in X_1 \rbrace ,\\ X_1 \odot A_1&=\lbrace a \in A \mid \exists a_1 \in A_1 : a_1 \twoheadrightarrow a \in X_1 \rbrace \end{align*}$$$$for any $$X_1, X_2 \in U(X), A_1 \in U(A)$$. In [17], Rump proved $$U(X)$$ is a logical quantale.…”

“…Let ( 𝑋 𝐴, ↠, ↣) be a quantum B-module, and let 𝑈(𝑋) and 𝑈(𝐴) be upper sets of 𝑋 and 𝐴, respectively. It was shown in [23] that 𝑈(𝑋) 𝑈(𝐴) is a quantale module with respect to set-theoretic union and the multiplications…”

“…Further, Xia and Zhao constructed the concrete forms of the least and the largest standard completion of quantum B-algebras [22], where the least standard completion is just the completion defined in [15]. Recently, Zhang and Rump introduced the notion of quantum B-modules over quantum B-algebras, and gave an explicit construction of the injective envelope of a quantum B-module which is just a quantale module [23].…”