On satisfiability of the \(C'(1/3)\) and \(C(4)\) conditions for special homogeneous semigroups with defining word-powers (Q1337872)

A semigroup \(\Pi\) is called special homogeneous if it is defined on some alphabet \(a_1,\dots,a_n\) by a system of defining relations \(R_1=1,\dots,R_m=1\), and \(|R_i|=|R_j|\) for all \(i\), \(f=j,\dots,m\), where \(|W|\) is the length of a word \(W\). The author considers only the case in which \(\Pi\) is a group. A word \(X\) is called a cusp relative to the set \({\mathfrak R}=\{R_1,\dots,R_m\}\) of defining words if \(\mathfrak R\) contains different words \(R_i\) and \(R_j\) of the forms \(R_i=XA\) and \(R_j=XB\). Condition \(C'(\lambda)\): if \(R\in{\mathfrak R}\), \(R=XA\), where \(X\) is a cusp, then \(|X|<\lambda|R|\) (\(\lambda\) is a positive real number). Condition \(C(p)\): if \(R\in {\mathfrak R}\), \(R=X_1\dots X_k\), where \(X_1,\dots, X_k\) are cusps, then \(k\geq p\) (\(p\) is a positive integer). It is clear, that \(C'(\lambda)\) with \(\lambda\leq 1/(p -1)\) implies the condition \(C(p)\). Let \(\Pi_r\) denote a special homogeneous semigroup in which each \(R_i\) of \(\mathfrak R\) is of the form \(U^{\alpha_i}_i\) with \(\alpha_i\geq r\), where \(U_i\) is a primitive word in \(\Pi_r\). The following two theorems are proved. The set \(\mathfrak R\) of the semigroup \(\Pi_5\) satisfies the condition \(C'(1/3)\). The set \(\mathfrak R\) of the semigroup \(\Pi_4\) satisfies the condition \(C(4)\). The author presents an example of \(\Pi_4\) in which \(C'(1/3)\) is not satisfied and an example of \(\Pi_3\) in which \(C(4)\) is not satisfied. As an application of the second theorem, there is given a very simple algorithm for deciding the word problem in \(\Pi_4\).