## 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 $$y\cdot x=x)$$.

### MSC:

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