NON-REGULAR SEMIGROUPS WHICH ARE AMALGAMATION BASES (New contact points of algebraic systems, logics, languages, and computer sciences)
邦孝 庄司 · Kyoto University Research Information Repository (Kyoto University) · 2015
In this paper, we study non-regular semigroups which are amalgamation basess for finite semigroups or for all semigroups.1 Semigroup amalgamation bases Definition.Let $\mathcal{A}$ be the class of finite semigroups or the class of all semigroups.Let $S,$ $T,$ $U$ be semigroups in $\mathcal{A}$ such that $U$ is a subsemigroup of $S$ and $T$ in common.Then a triple $[S, T;U]$ is called an amalgam of semigroups $S,$ $T$ with $U$ as a core in $\mathcal{A}$ .An amalgam $[S, T;U]$ of $\mathcal{A}$ is called to be weakly embedable in $\mathcal{A}$ if there exist a semigroup $K$ belonging to $\mathcal{A}$ and monomorphisms $\xi_{1}$ : $Sarrow K,$ $\xi_{1}$ : $Tarrow K$ such that the restrictions to $U$ of $\xi_{1}$ and $\xi_{2}$ are equal to each other $($ that $is, \xi_{1}(S)\cap\xi_{2}(T)\supseteq\xi_{1}(U))$ .An amalgam $[S, T;U]$ of $\mathcal{A}$ is called to be strongly embeddable in $\mathcal{A}$ if $\xi_{1}(S)\cap\xi_{2}(T)=\xi_{1}(U)$ .A semigroup $U$ in $\mathcal{A}$ is amalgamation base [resp.weak amalgamation $base|$ if any amalgam with a core $U$ in $\mathcal{A}$ is strongly embeddable [resp.weakly embeddable] in $\mathcal{A}.$We have the following results which will be used later.Result $1$ [ $[4]$ , Theorem 12].Any finite semigroup $U$ is an amalgamation base for finite semigroups if and only if $U$ is a weak amalgamation base for finite semigroups [for all semigroup] Result $2$ [ $[6]$ , Theorem 1].If a finite semigroup $U$ is an amalgamation base for finite semigroups, then all $\mathcal{J}$ -classes of $U$ form a chain.$*$ This is an absrtact and the paper will appear elsewhere.