Representations in varieties of regular involution bands. (English) Zbl 0731.18001

Koubek, V.

Commentat. Math. Univ. Carol. 31, No. 2, 201-211 (1990).

A regular involution band is a triple \((B,\cdot,+)\), where (B,\(\cdot)\) is a band (i.e. idempotent semigroup) and \(+\) is a unary operation on B satisfying the identities: \(x^{++}=x\), \(x\cdot x^+\cdot x=x\). For a variety \({\mathcal V}\) of bands denote by \({\mathcal R}{\mathcal I}{\mathcal V}\) the variety of all regular involution bands \((B,\cdot,+)\) for which (B,\(\cdot)\) belongs to \({\mathcal V}.\)
A category \({\mathcal K}\) is said to be universal if any category of algebras can be fully embedded into \({\mathcal K}\). No variety of bands is universal. It is proved that if either \({\mathcal V}\) is the variety of left normal bands (i.e. satisfies \(x\cdot y\cdot z=x\cdot z\cdot y)\) or \({\mathcal V}\) is the variety of right normal bands (i.e. satisfies \(y\cdot z\cdot x=z\cdot y\cdot x)\) then \({\mathcal R}{\mathcal I}{\mathcal V}\) is universal. Since universality is an increasing property, this gives: Let \({\mathcal V}\) be a variety of bands. Then \({\mathcal R}{\mathcal I}{\mathcal V}\) is universal if and only if \({\mathcal V}\) is a non-trivial variety distinct from the following three varieties: semilattices (i.e. commutative bands), left zero- semigroups (i.e. satisfying \(x\cdot y=x)\), right zero-semigroups (i.e. satisfying \(y\cdot x=x)\).

Reviewer: M.Demlová (Praha)

MSC:

18B15	Embedding theorems, universal categories
20M07	Varieties and pseudovarieties of semigroups
20M30	Representation of semigroups; actions of semigroups on sets

Keywords:

universal category; regular involution band; variety of bands