Partition monoids and embeddings in 2-generator regular \(*\)-semigroups.
In the introduction the author has written that ``it was shown in the paper \textit{D. B. McAlister, J. B. Stephen, A. S. Vernitski} [Proc. Edinb. Math. Soc., II. Ser. 45, No. 1, 1-4 (2002; Zbl 1013.20059)] that the full transformation semigroup \(\mathcal T_n\) (which is minimally generated by 3 elements) embeds in 2-generator subsemigroup of \(\mathcal T_{n+1}\). Coupled with Cayley's Theorem for semigroups, this shows that any finite semigroup embeds in a finite 2-generator semigroup. Analogous result for infinite transformation semigroups goes back to \textit{W. Sierpiński} [Fundam. Math. 24, 209-212 (1935; Zbl 0011.10607)].'' Due to this result any countable semigroup embeds in a 2-generator semigroup. In the current paper the author investigates the analogous problems for partition monoids, a naturally occurring class of \(*\)-regular semigroups that contains both the full transformation semigroups and symmetric inverse semigroups. Now the author proves that (1) the partition monoid \(\mathcal P_n\) (which is minimally generated by 4 elements) embeds in 3-generator subsemigroup of \(\mathcal P_{n+1}\); (2) the partition monoid \(\mathcal P_n\) does not embed in 2-generator subsemigroup of \(\mathcal P_{n+1}\) if \(n\geq 3\); (3) the partition monoid \(\mathcal P_n\) embeds in 2-generator subsemigroup of \(\mathcal P_{n+4}\). A consequence of (3) is (4) every finite semigroup embeds in a finite 2-generated \(*\)-regular semigroup. In conclusion of the introduction the author states a number of open questions.
scientific article; zbMATH DE number 6384190
scientific article