Idempotents, regular elements and sequences from finite semigroups. (English) Zbl 0871.20051
The paper proves a combinatorial result, the likes of which have proved useful in finite semigroup theory, that if \(n\) is the number of non-idempotent elements of a finite semigroup \(S\) then any product of length \(2^n\) has an idempotent factor (regarding the product as a word). The proof is a straightforward induction argument and examples which exploit properties of Zimin words show the bound of \(2^n\) is best possible.
From this they calculate explicitly the integer \(b(n)\) such that for every semigroup of \(n\) elements any product of length \(b(n)\) has an idempotent factor. The value of \(b(n)\) is of the order of \(4^{n/3}\), the precise value depends on \(n\pmod 3\). The proofs allow similar results to be derived for other semigroup classes and element types.
From this they calculate explicitly the integer \(b(n)\) such that for every semigroup of \(n\) elements any product of length \(b(n)\) has an idempotent factor. The value of \(b(n)\) is of the order of \(4^{n/3}\), the precise value depends on \(n\pmod 3\). The proofs allow similar results to be derived for other semigroup classes and element types.
Reviewer: P.M.Higgins (Colchester)
MSC:
| 20M05 | Free semigroups, generators and relations, word problems |
Keywords:
regular elements; numbers of nonidempotent elements; finite semigroups; idempotent factors; Zimin wordsReferences:
| [1] | Ash, C. J., Finite semigroups with commuting idempotents, J. Austral. Math. Soc. Ser. A, 43, 81-90 (1987) · Zbl 0634.20032 |
| [2] | Clifford, A. H.; Preston, G. B., The algebraic theory of semigroups, (Math. Surveys, Vols. I, II (1961), Amer. Math. Soc: Amer. Math. Soc Providence I), 1967, No. 7 · Zbl 0178.01203 |
| [3] | Easdown, D., Biordered sets of eventually regular semigroups, (Proc. London Math. Soc., 49 (1984)), 483-503, (3) · Zbl 0568.20059 |
| [4] | Gillam, D. W.H.; Hall, T. E.; Williams, N. H., On finite semigroups and idempotents, Bull. London Math. Soc., 4, 143-144 (1972) · Zbl 0252.20064 |
| [5] | Hall, T. E., On regular semigroups, J. Algebra, 24, 1-24 (1973) · Zbl 0262.20074 |
| [6] | Higgins, P., Epimorphisms, permutation identities and finite semigroups, Semigroup Forum, 29, 87-97 (1984) · Zbl 0546.20051 |
| [7] | Kaufman, A. M., Sequentially annihilating sums of associative systems, Uch. zap. Leningr. gos. ped. in-ta im Gercena, 86, 145-166 (1949), (in Russian) |
| [8] | Kharlampovich, O.; Sapir, M., Algorithmic problems in varieties, Internat. J. Algebra and Comp., 5, 379-602 (1995) · Zbl 0837.08002 |
| [9] | Simon, I., Word Ramsey Theorems, (Bollobàs, B., Graph Theory and Combinatorics (1984), Academic Press: Academic Press London), 283-291 · Zbl 0548.20043 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
