On the problem of embedding elements of groups in subsemigroups (Q1326912)

Let \(G = \langle {\mathfrak A} \mid R = 1\;(R \in {\mathfrak R})\rangle\) be a finitely presented group, where \(\mathfrak R\) is closed with respect to cyclic shifts of the words of \(R\) and inversion in the free group. A word \(X\) is said to be a cusp (with respect to \(\mathfrak R\)) if \(\mathfrak R\) contains different \(R_ i\) and \(R_ j\) that are respectively of the form \(R_ i \eqcirc XP\) and \(R_ j \eqcirc XQ\). Without loss of generality, we can assume that each letter of the alphabet \(G\) is a cusp. Let \(\| W\|\) be the smallest number of nonempty cusps whose product is \(W\). The condition \(C(p)\) for \(G\) can be stated in the following form: for an \(R \in {\mathfrak R}\), \(\| R\| > p\) (where \(p\) is a natural number). A word \(P\) is said to be \(j\)-residual (with respect to \(\mathfrak R\)) if some \(R \in {\mathfrak R}\) is of the form \(R \eqcirc PX_ 1 \dots X_ j\), where \(X_ 1,\dots, X_ j\) are cusps. A group \(G\) satisfies the condition \(E(3)\) if there is no 3-residual word in the positive alphabet of the group \(G\). For all groups that satisfy the conditions \(C(G)\) and \(E(3)\) the author solves the membership problem of elements of the group \(G\) to the subsemigroup \(S\) generated by the positive alphabet of the group \(G\). The author also investigates the problem of determining a system of defining relations for \(S\).