×

On satisfiability of non-trivial identities in finitely generated semigroups. (Russian) Zbl 0683.20049

Let A be an alphabet of power \(\geq 2\). A word X in A is said to be an \(\alpha\)-word, where \(\alpha\) is a positive real number, if X has a subword of the kind \(Y^{[\alpha \cdot \ell (Y)]+6}\) (here \(\ell (Y)\) is the length of Y and [\(\beta\) ] is the greatest integer in \(\beta)\). Theorem 1. Let \(\Pi\) be a semigroup which is defined by a set \(\{A_ i=B_ i|\) \(i=1,2,...\}\) of defining relations in A such that all \(A_ i\), \(B_ i\) are \(\alpha\)-words for some \(\alpha\). Then \(\Pi\) satisfies no non-trivial identities. This theorem gives a simple method of construction of 2-generated periodic semigroups which satisfy no non- trivial identities. Theorem 2. For any \(k\geq 2\) there exists a k- generated semigroup without non-trivial identities such that all its (k- 1)-generated subsemigroups are nilpotent. This theorem strengthens a result of E. Golod [Izv. Akad. Nauk SSSR, Ser. Mat. 28, 273-276 (1964; Zbl 0215.39202)].
Reviewer: S.R.Kogalovskij

MSC:

20M07 Varieties and pseudovarieties of semigroups
20M05 Free semigroups, generators and relations, word problems

Citations:

Zbl 0215.39202
PDFBibTeX XMLCite
Full Text: EuDML