On distortion of subsemigroups (Q1596157)

Let \(S\) be an arbitrary semigroup, \(l\colon S\to\mathbb{Z}^+\) be some function. According to A.~Yu.~Ol'shanskij it is of interest under what conditions on \(l\) the semigroup \(S\) can be embedded into some semigroup \(H\) of a definite class so that \(l(s)\sim|s|_H\), where \(|s|_H\) is the length of the shortest word in \(H\) representing \(s\). The author finds an answer to this question when \(H\) is finitely generated. Also sufficient conditions on \(l\) are established for the case when \(H\) is finitely presented. The results obtained are similar to the corresponding theorems for groups [see \textit{A.~Yu.~Ol'shanskij}, Proc. special year in geometric group theory, Canberra 1996, Berlin, de~Gruyter, 281-291 (1999); Mat. Sb. 188, No.~1, 51-98 (1997; Zbl 0905.20020)].