The local and global varieties induced by nilpotent monoids. (English) Zbl 0607.20035

The wreath product MNIL*D, where MNIL is the variety of finite monoids (in the sense of Eilenberg) generated by nil monoids (i.e. such that \(x^ n=0\) for \(x\neq 1)\) and D is the variety of finite semigroups \(\{\) \(S: Se=e\) for all \(e\}\) is equal to \(LMNIL=\{S:\) eSe\(\in MNIL\) for \(e\in S\), \(e^ 2=e\}\), the variety of finite locally nil semigroups. This yields a method to decide whether a semigroup can be embedded into a wreath product \(N\circ d\), \(N\in MNIL\), \(d\in D\). - [Reviewer’s remark: The reviewer cannot see the advantage of broadly using traditional terms in non-traditional sense (besides, without advertising the reader).]
Reviewer: G.Pollák


20M07 Varieties and pseudovarieties of semigroups
20M05 Free semigroups, generators and relations, word problems
20M35 Semigroups in automata theory, linguistics, etc.
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