Certain embedding problems of semigroups, II
Reikichi Yoshida · Proceedings of the Japan Academy Series A Mathematical Sciences · 1965
In this paper the author will discuss the problems 1, 3 presented by T. Tamura and N. Graham _4.The terminology and the numbers of formulas in the previous paper will be used here without definitions./P denote the left right translation semigroup of a semigroup S. The necessary and sufficient condition so that S is embeddable in the right-sided way was given by Theorem 3 in 4. But we can find subsemigroups A of Theorem 3 in several ways.We wish to rewrite Theorem 3.Let C be the set of all left translations of S such that p-p for all right translations p of S, and D the set of all left translations of S which has a linked right translation of S. If we set A-C D, then we can prove easily that A becomes a semigroup containing the identical mapping _1 and the inner let translation semigroup -//0.If 2 and p are linked, we write (LK)p.As in 4, we have Lemma a.If v e 4 P, then it follows that v(LK).Moreover by Lemma a, Lemma b.If S is embeddable in the mixed way, then D--4.