Construction of finite commutative $z$-semigroups
Miyuki Yamada · Proceedings of the Japan Academy Series A Mathematical Sciences · 1964
1. Introduction.As defined by Tamura 4, a semigroup is called a z-semigroup if it has a zero element, 0, but has no idem- potent except 0. In particular, for a finite commutative semigroup S it is easily seen that S is a z-semigroup if and only if it satisfies the following two conditions:(1) S has a zero element 0 and (2) SS .S={0} for some positive integer p.If S\S is non-empty, every element of S\S is called a prime element of S.In the case of p=l or p=2, S satisfies the following (3) z={0} or (4) xy=O for any x, yeS, respectively.Such a semigroup S is called a trivial z-semigroup or a null semigroup, corresponding to p=l or p--2.Now, the problem of construction of finite commutative z-semi- groups occupies an important part in the problem of construction of finite commutative semigroups.In this paper, we shall deal with this problem and present a method of constructing all possible com- mutative z-semigroups of a given order.The proofs are omitted and will be given in detail elsewhere. )