Note on a semigroup having no proper subsemigroup
Takayuki Tamura · Proceedings of the Japan Academy Series A Mathematical Sciences · 1961
In the previous paper [1 we determined the structure ofsemigroup and added that a finite semigroup of orders>2 which con- tains no proper subsemigroup is a cyclic group of prime order.Further we noticed, without proof, that this holds even if the condition "finiteness" is excluded.In the present note we shall prove the theorem without using the result of -semigroup.Theorem.A semigroup of order')2 which has no proper sub- semigroup is a cyclic group of prime order.Let S be a semigroup of order>2 which has no proper subsemi- group, and let a and b be arbitrary distinct elements of S. Then we see that S is generated by a and b.First we must prove that S contains at least a non-idempotent element.For this purpose we may show that an idempotent semigroup M of order>2 generated by the two distinct elements a and b has at least one proper subsemigroup.Now, F denotes the free idempotent semigroup generated by a and b.M is given as a suitable ".) factor semigroup of F. Fortunately it is easily proved a) that F is a semigroup of order 6 which consists of a, b, ab, ha, aba, bah.