Subalgebra systems of idempotent entropic algebras
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.
idempotent-entropic operator semilattice
scientific article; zbMATH DE number 4083697
scientific article