Semigroups whose arbitrary subsets containing a definite element are subsemigroups
Morio Sasaki · Proceedings of the Japan Academy Series A Mathematical Sciences · 1963
Consider a semigroup S satisfying the following condition: Any subset of S which contains a definite element e is a subsemi- group of S.A semigroup S is called a fl*-semigroup if S satisfies the above condition.For example semigroups of order 2, fl-semigroups [4 x) and Rdei's semigroups are all fl*-semigroups, where by a Rdei's semigroup we mean a semigroup satisfying the condition that any non-empty subset is a subsemigroup [2J. ) 2. Immediately we have that a homomorphic image of S is a *-semigroup and any subset of S which contains e is also a fl*semigroup.Putting now T--{xeS; x=x}, U--{xeS; x=e, xe, ex--xe--e}, and V=[xeS; x=e, x#e, ex=xe=x}, it follows that V has at most one element and S= T+ U--V (disjoint class-sum).