Internalizing equality in Boolean algebras
type: Item
instance of: scholarly article
title: Internalizing equality in Boolean algebras
zbMATH Open document ID: 1011.06014
author: Desmond Fearnley-Sander
author: T. E. Stokes
published in: Algebra Universalis
publication date: 2003-04-23T00:00:00Z
review text: An equality algebra is a universal algebra \(A\) together with a semilattice \((L,\cdot ,1)\) and a binary function \(=_i : A\times A \rightarrow L\) satisfying the reflexivity rule \((x=_i x)=1\) and the replacement rule \((x=_i y)f(x)=(x=_i y)f(y)\) for all functions \(f:A\rightarrow L\) derived from the operations on \(A\), the semilattice operation and \(=_i\). The most important case the authors consider is that one in which the Boolean ring \(R\) has \(=_i\) taking values in the semilattice \((R,\cdot ,1)\). This is called an EB-ring. It is shown that the variety of EB-rings is equivalent to the variety of modal rings (Boolean rings with an interior operation). Varying the strength and exact nature of the replacement property corresponds to selecting from a number of natural varieties of modal rings.
reviewed by: Ivan Chajda
Mathematics Subject Classification ID: 06E25
Mathematics Subject Classification ID: 03G25
Mathematics Subject Classification ID: 03B45
zbMATH DE Number: 1899932
zbMATH Keywords: Boolean ring
zbMATH Keywords: modal ring
zbMATH Keywords: equality algebra
zbMATH Keywords: replacement property
MaRDI profile type: Publication
full work available at URL: s000120050152
OpenAlex ID: W1994809543
DOI: 10.1007/S000120050152
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description: scientific article; zbMATH DE number 1899932