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Semigroup presentations. (English) Zbl 0981.20047

Shum, Kar Ping (ed.) et al., Semigroups. Papers from the international conference on semigroups and its related topics, Kunming, China, August 18-23, 1995. Singapore: Springer. 178-187 (1998).
Some recent results on semigroup presentations are presented. These results concern two directions: (1) given a known semigroup find a presentation of the semigroup; (2) given a presentation give a description of the semigroup with this presentation.
The following open problem on the rank of a semigroup is discussed (recall that if \(S\) is a finitely generated semigroup then \(\text{rank}(S)=\min\{|X|: X\subset S\), \(\langle X\rangle= S\}\)): Suppose that \(M\) is a finite monoid of rank \(r\) and let \(S\) be a finite Rees matrix semigroup \(S={\mathcal M}^0[M;I,\Lambda,P]\). Find the rank of \(S\). Some remarkable results concerning this problem have been obtained by N. Ruškuc [Math. Proc. Camb. Philos. Soc. 116, No. 2, 325-338 (1994; Zbl 0817.20062)].
Related with (2), starting with some classical presentations of groups find descriptions of the semigroups admitting the same presentations. Results obtained by C. M. Campbell, E. F. Robertson, N. Ruškuc and R. M. Thomas [Proc. R. Soc. Edinb., Sect. A 125, No. 5, 1063-1075 (1995; Zbl 0840.20058)] are presented. The semigroups having as presentations \[ \begin{split}\langle a_1,\dots,a_n\mid a^3_i= a_i,\;(a_ja_{j+1})^3=a^2_j,\\ a_ka_l=a_la_k\;(1\leq i\leq n,\;1\leq j\leq n-1,\;1\leq k\leq l- 2\leq n-2)\rangle,\end{split} \] respectively \[ \langle a_1,\dots,a_n\mid a^4_i=a_i,\;(a_ja_k)^2=a^3_j,\;1\leq i\leq n,\;1\leq j\leq k-1\leq n-1\rangle \] are described.
For the entire collection see [Zbl 0946.00017].

MSC:

20M05 Free semigroups, generators and relations, word problems
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