Subalgebra systems of idempotent entropic algebras
type: Item
instance of: scholarly article
title: Subalgebra systems of idempotent entropic algebras
zbMATH Open document ID: 0663.08001
DOI: 10.1016/0021-8693(89)90197-X
author: Anna Romanowska
author: J. D. H. Smith
published in: Journal of Algebra
publication date: 1989-01-01T00:00:00Z
review text: An algebra \(<A,\Omega >\) is said to be entropic if, for each n-ary operation \(\omega\) and \(n'\)-ary operation \(\omega '\), the action \(\omega\) : \(<A^ n,\omega '>\to <A,\omega '>\) is a homomorphism of \(\omega '\)- algebras. An idempotent-entropic operator semilattice (or IEO- semilattice) \(<B,+,\Omega >\) is an idempotent entropic algebra \(<B,\Omega >\) with a semilattice structure \(<B,+>\) such that \(\Omega\) is distributive over \(+\), i.e. for each n-ary \(\omega\in \Omega\), \(x_ 1,...,x_ n,x'\in B\), and \(1\leq j\leq n\), \(x_ 1,...(x_ j+x_ j')...x_ n\omega =x_ 1...x_ j...x_ n\omega +x_ 1...x_ j'...x_ n\omega.\) In the present paper the subalgebra systems of idempotent entropic algebras are considered as IEO-semilattices. The authors discuss the relationship between the system of finitely generated subalgebras and that of all subalgebras. The obtained results enable to derive some fundamental properties of the variety B of all IEO- semilattices whose \(\Omega\)-reducts satisfy a given set of regular linear identities. Especially the free IEO-semilattices in B are described and the Representation Theorem for IEO-semilattices from B is proved. Examples of idempotent entropic algebras include sets, semilattices, normal bands, and vector spaces under the formation of centroids. Corresponding examples of IEO-semilattices include semilattices, meet- distributive bisemilattices, semilattice-normal semirings, and some semilattice-ordered groupoids.
Mathematics Subject Classification ID: 08A30
Mathematics Subject Classification ID: 06A12
zbMATH DE Number: 4083697
zbMATH Keywords: idempotent-entropic operator semilattice
zbMATH Keywords: idempotent entropic algebra
zbMATH Keywords: subalgebra systems
reviewed by: Jaromír Duda
MaRDI profile type: Publication
cites work: Q5331549
cites work: Q5515373
cites work: Q3898534
cites work: Q4126563
cites work: Q5639839
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cites work: Bisemilattices of subsemilattices
cites work: On varieties closed under the construction of power algebras
cites work: Étude algébrique de la notion de moyenne
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description: scientific article; zbMATH DE number 4083697