Automatic subsemigroups of free products. (Q2482061)

The notion of automatic group recently has been extended to semigroups and the basic properties of this new class of semigroups have been established by \textit{C. M. Campbell}, \textit{E. F. Robertson}, \textit{N. Ruškuc} and \textit{R. M. Thomas} [Theor. Comput. Sci. 250, No. 1-2, 365-391 (2001; Zbl 0987.20033)]. Given a finite alphabet \(A\), the author denotes by \(A^+\) the free semigroup generated by \(A\) consisting of finite sequences of elements of \(A\) under the concatenation, and by \(A^*\) the free monoid generated by \(A\) consisting of \(A^+\) together with the empty word \(\varepsilon\). Let the set \(A(2,\$)=((A\cup\{\$\})\times(A\cup\{\$\}))\setminus\{\$,\$\}\), where \(\$\) is a symbol not in \(A\). The function \(\delta_A\colon A^*\times A^*\to A(2,\$)^*\) is defined by \[ (a_1\cdots a_m,b_1\cdots b_n)\delta_A=\begin{cases}\varepsilon\quad &\text{if }0=m=n,\\ (a_1,b_1)\cdots(a_m,b_m)\quad &\text{if }0<m=n,\\ (a_1,b_1)\cdots(a_m,b_m)(\$,b_{m+1})\cdots(\$,b_n)\quad &\text{if }0\leq m<n,\\ (a_1,b_1)\cdots(a_n,b_n)(a_{n+1},\$)\cdots(a_m,\$)\quad&\text{if }m>n\geq 0.\end{cases} \] Let \(S\) be a semigroup and \(A\) a finite generating set for \(S\) with respect to \(\psi\colon A^+\to S\). The pair \((A,L)\) is an automatic structure for \(S\) (with respect to \(\psi\)) if the following three properties hold: (1) \(L\) is a regular subset of \(A^+\) and \(L\psi=S\); (2) \(L_==\{(\alpha,\beta):\alpha,\beta\in L,\;\alpha=\beta\}\delta_A\) is regular in \(A(2,\$)^+\); (3) \(L_a=\{(\alpha,\beta):\alpha,\beta\in L,\ \alpha a=\beta\}\delta_A\) is regular in \(A(2,\$)^+\) for each \(a\in A\). A semigroup is automatic if it has an automatic structure. The main result is the following theorem. Let \(S\) be a free product of finitely many semigroups \(S=S_1*\cdots*S_n*T_1\cdots*T_m\), where \(T_1,\dots,T_m\) are free semigroups on finite sets \(Y_1,\dots,Y_m\), respectively. Let \(H=(t_1,\dots,t_l)\) be a subsemigroup of \(S\), where \(t_1,\dots,t_l\in S\setminus(S_1\cup\cdots\cup S_n)\). Then \(H\) is an automatic semigroup.