Miyuki Yamada

Orthodox semigroups whose idempotents satisfy a certain identity

An orthodox semigroup S is called a left [right] inverse semigroup if the set of idempotents of S satisfies the identity xyx=xy [xyx=yx]. Bisimple left [right] inverse semigroups have been studied by Venkatesan [6]. In this paper, we clarify the structure of general left [right] inverse semigroups. Further, we also investigate the structure of orthodox semigroups whose idempotents satisfy the identity xyxzx=xyzx. In particular, it is shown that the set of idempotents of an orthodox semigroup S satisfies xyxzx=xyzx if and only if S is isomorphic to a subdirect product of a left inverse semigroup and a right inverse semigroup.

P-systems in regular semigroups

In this paper, firstly it is shown that a regular semigroup S becomes a regular *-semigroup (in the sense of [1]) if and only if S has a certain subset called a p-system. Secondly, all the normal *-bands are completely described in terms of rectangular *-bands (square bands) and transitive systems of homomorphisms of rectangular *-bands. Further, it is shown that an orthodox semigroup S becomes a regular *-semigroup if there is a p-system F of the band ES of idempotents of S such that F∋e, ES∋t, e≥t imply t∈F. By using this result, it is also shown that F is a p-system of a generalized inverse semigroup S if and only if F is a p-system of FS.

Note on a certain class of orthodox semigroups

This is a continuation and also a supplement of the previous papers [5], [6] and [8] concerning orthodox semigroups1). In [8], it has been shown that a quasi-inverse semigroup is isomorphic to a subdirect product of a left inverse semigroup and a right inverse semigroup. In this paper, we present a structure theorem for quasi-inverse semigroups and some relevant matters.

O-simple strictly regular semigroups

In the previous paper [6], it has been proved that a semigroup S is strictly regular if and only if S is isomorphic to a quasi-direct product EX Λ of a band E and an inverse semigroup Λ. The main purpose of this paper is to present the following results and some relevant matters:(1) A quasi-direct product EX Λ of a band E and an inverse semigroup Λ is simple [bisimple] if and only if Λ is simple [bisimple], and (2) in case where EX Λ has a zero element, EX Λ is O-simple [O-bisimple] if and only if Λ is O-simple [O-bisimple]. Any notation and terminology should be referred to [1], [5] and [6], unless otherwise stated.

Note on exclusive semigroups

In the SEMIGROUP FORUM, Vol. 1, No. 1, B. M. Schein proposed the following problem:Describe the structure of semigroups S such that for every a,b,c∈S, abc=ab, bc or ac. At present, we shall call such a semigroup S anexclusivesemigroup. Recently, the author heard that the structure of commutative exclusive semigroups was completely determined by T. Tamura [3]. In this paper, we deal with exclusive semigroups which are not necessarily commutative. The paper is divided into three sections. At first, the structure of exclusive semigroups whose idempotents form a rectangular band will be clarified. Next, we shall investigate a certain class of exclusive semigroups called “exclusive homobands”. Especially, in the final section we shall deal with medial exclusive homobands and show how to construct them. The proofs are omitted and will be given in detail elsewhere.

On commutativity of a semigroup which is a semilattice of commutative semigroups

Let P,(G) and P,(G) be abstract properties pertaining to commutative semigroups G in the sense of Cohn [3]. P,(G) is said to be weaker than or equal to P,(G) and denoted by P,(G) 3 P,(G) if and only if, for any commutative semigroup S, P,(G) is satisfied by S (i.e., P,(S) is true) whenever P,(G) is satisfied by S. If Pi(G) 2 P,(G) and P,(G) 3 P,(G), then P,(G) and P,(G) are said to be equivalent and denoted by P,(G) = P,(G). If P,(G) P,(G), we regard P,(G) and P,(G) as the same property. When S is a semigroup which is a semilattice of commutative semi-groups S, , 5 E x, S is not necessarily commutative. However, there is an abstract property P(G) pertaining to commutative semigroups G, such that, any semigroup which is a semilattice of commutative semigroups with P(G) is commutative. Such an abstract property P(G) is called a fully c-invariant property (abbrev., f.c.i.-property). For example, it is well-known (e.g., see Clifford [I]) that the property P(G), “G is a group”, is an f.c.i.-property. There is no greatest (i.e., weakest) f.c.i.-property with respect to the ordering relation defined above, but there is a maximal f.c.i-property. Further, a maximal f.c.i.property is not unique. The main purpose of this paper is to obtain maximal f.c.i.-properties, and some relevant results.

D*-simple inverse semigroups

Let ~ s be the Greens D-relation on a semigroup S, and ~s* the congruence on S generated by ~ s , that is, the least congruence (on S) containing ~ s I f ~s* is the universal congruence, that is, if ~s* satisfies a~*sb for all a, b E S, then S is said to be D*-simple. According to Hall [1], the least semilattice congruence ~s on a regular semigroup S coincides with 2s*; that is, ~?s = ~s*. From this fact, it follows from Petrich [6] that the ~Us-relation , U s on a regular semigroup S in the sense of [6] also coincides with ~s*. We shall call ~s* the D*-relation on S. Now, let S be a regular semigroup. There exist a semilattice A and a regular semigroup Sa for each A ~ A such that S is a semilattice A of the regular semigroups S a and each S a is an ~?s-class. In this case, each Sa is semilattice indecom: posable (S-indecomposable) and hence S a is D*-simple. Tha t is, a regular [orthodox, inverse] semigroup S is a semilattice of D*-simple regular [orthodox , inverse] semigroups. Conversely, let A be a semilattice and Sa an orthodox semigroup for each A ~ A. Let S(o) be a semilattice composition of (Sa:. A t A}; that is, S(o) is a semigr0up such that