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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

MSC:

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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References:

[1] H. STRAUBING, The Variety Generated by Finite Nilpotent Monoids, Semigroup Forum, Vol. 24, 1982, pp. 25-38. Zbl0503.20024 MR645701 · Zbl 0503.20024 · doi:10.1007/BF02572753
[2] D. THÉRIEN and A. WEISS, Graph Congruences and Wreath Products, J. P. Ap. Alg., Vol. 36, 1985, pp. 205-215. Zbl0559.20042 MR787173 · Zbl 0559.20042 · doi:10.1016/0022-4049(85)90071-4
[3] A. WEISS and D. THÉRIEN, Varieties of Finite Categories, R.A.I.R.O. Informatique Théorique, Vol. 20, n^\circ 3, 1986, pp. 357-366. Zbl0608.18002 MR894719 · Zbl 0608.18002
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