Certain embedding problems of semigroups
Takayuki Tamura, Nancy Graham · Proceedings of the Japan Academy Series A Mathematical Sciences · 1964
1.By a left translation of a semigroup S we mean a trans- formation of S, xx, satisfying (xy)-(x)y, for all x, y in S. A right translation of S is a transformation p satisfying (xy)p-x(yp), for all x, y in S. A left translation and a right translation p are said to be linked if x(y)-(xp)y, for all x, y in S. We note that for each a in S, the transformation defined by x-ax, for all x in S, is a left translation of S, the transformation p defined by -xa, for all x in S, is a right translation of S, and and p are linked.We call a an inner left translation of S, p an inner right translation of S. A semigroup S is said to be weakly reductive if, for any a, b in S, ax-bx and xa-xb, for all x in S, imply a--b.It was proved in 1] that a weakly reductive semigroup S can be embedded into a semigroup T so that(1) S is an ideal of T, (2) every left translation of S is induced by some inner left translation of T, and every right translation of S is induced by some inner right translation of T, if and only if