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Universal varieties of semigroups. (English) Zbl 0549.20038

This paper continues the investigation pioneered by Z. Hedrlín and J. Lambek [in J. Algebra 11, 195-212 (1969; Zbl 0206.02505)], but restricts its attention to varieties of semigroups. A category \(V\) is called universal (or binding) if every category of algebras is isomorphic to a full subcategory of it. Since a variety, i.e., an equationally defined class of algebras, is a category, one may consider universal varieties. Universal varieties of semigroups are characterized by theorem: A semigroup variety \(V\) is universal iff (i) it is definable by identities \(p=q\) such that \(p\) and \(q\) have the same total degree in each variable, and (ii) \(V\) satisfies no equation of the form \((xy)^ n=x^ ny^ n\) for \(n>1\). A consequence derived from this is that every universal variety of semigroups must contain the variety of commutative semigroups. The main results are proved by constructing a full embedding of the category \(G\) of connected undirected graphs into any such \(V\) and then applying the result [Z. Hedrlín and A. Pultr, Monatsh. Math. 68, 421-425 (1964; Zbl 0139.24802)] that \(G\) is universal.
Reviewer: W.R.Nico

MSC:

20M07 Varieties and pseudovarieties of semigroups
18B15 Embedding theorems, universal categories
20M50 Connections of semigroups with homological algebra and category theory
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