Topology lattices of unary algebras
type: Item
instance of: scholarly article
title: Topology lattices of unary algebras
zbMATH Open document ID: 1011.08007
published in: Russian Mathematical Surveys
publication date: 2003-06-02T00:00:00Z
Mathematics Subject Classification ID: 08A60
Mathematics Subject Classification ID: 54A10
Mathematics Subject Classification ID: 06B23
zbMATH DE Number: 1681667
zbMATH Keywords: lattice of topologies
zbMATH Keywords: congruence lattice
zbMATH Keywords: unary algebra
zbMATH Keywords: principal filter
zbMATH Keywords: unar
author: A. V. Kartashova
MaRDI profile type: Publication
full work available at URL: rm2000v055n06abeh000345
OpenAlex ID: W2025993798
DOI: 10.1070/RM2000V055N06ABEH000345
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review text: Let \(A=(A,F)\) be an algebra. A topology on the set \(A\) in which all the operations in \(F\) are continuous is called a topology on algebra \(A\). It is obvious that the topologies on algebra \(A\) form a complete lattice, which is denoted by \(T(A)\). In this paper the following main results are presented.NEWLINENEWLINENEWLINE1) The dual lattice of the congruence lattice of an algebra can be embedded in the topology lattice of this algebra.NEWLINENEWLINENEWLINE2) Let \(A\) be a unary algebra and let \(B\) be a subalgebra of \(A\). Then \(T(B)\) can be embedded as a principal filter in the lattice \(T(A)\).NEWLINENEWLINENEWLINE3) Let \(A\) be a unar, i.e. an algebra with exactly one unary operation. Then the lattice \(T(A)\) is isomorphic to \(\text{Con}(A)\) iff \(A\) is a cycle.NEWLINENEWLINENEWLINE4) Unars \(A\) are characterized for which \(T(A)\) is a modular lattice, a chain, a lattice with complements or with pseudocomplements, respectively.
reviewed by: Yuri Movsisyan
description: scientific article; zbMATH DE number 1681667