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

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 \(y\cdot x=x)\).


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