Semigroups and their subsemigroup lattices
Takayuki Tamura · Pacific Journal of Mathematics · 1963
1. Introduction.Let S be a semigroup of order at least 2, and L(S) be the system of all subsemigroups of S. Generally L(S), including the empty subset, is a lattice with respect to inclusion.L(S) is called the subsemigroup lattice of S. A semigroup S contains at least one nonempty subsemigroup besides S itself.In the previous paper [4], as the first step towards the investigation of the structure of S with a given type of L(S), we determined all the /'-semigroups, 1 namely, the semigroups S's in which L(S)'s are chains.In the present paper we shall define Γ*-semigroups as generalization of Γ-semigroups and shall obtain all the types of /^-semigroups except for infinite simple Γ* -groups.Since all the semigroups of order 2 are Γ* -semigroups, we shall treat non-trivial /^-semigroups, namely, those of order Ξ> 3 in the discussion below.First, in §2 we shall prove that Γ*-semigroups of order ^ 3 are unipotent, i.e., having a unique idempotent, and that they are periodic; and hence a /"^-semigroup is determined by a group and a ^-semigroup, i.e., a unipotent semigroup with zero.Accordingly, in §3 we shall determine all the types of Γ* -^-semigroups which will have to be of order <5; in §4 we shall treat solvable Γ*-groups and prove that finite /^*-groups or non-simple /""-groups are solvable; finally in § 5, unipotent Γ*-semigroups which are neither groups nor Z-semigroups will be discussed.It is interesting that there are no infinite unipotent Z 1 *-semigroups except groups.For convenience, the results from the paper [4] are stated as follows:LEMMA 1.1.A semigroup is a Γ-semigroup if and only if it has one of the following types.2Except for (1.3) they are all cyclic semigroups, i.e., semigroups generated by an element d.We show defining relations below.(1.1) Z-semigroups:(1.1.1)