Kunitaka Shoji

Absolute flatness of the full transformation semigroups

In [S], T. E. Hall showed that every semigroup with the strong representation extension property is an amalgamation base in the class of all semigroups and that a semigroup S has the strong representation extension property if and only if S has both the representation extension property and the free representation extension property. J. M. Howie [7] and S. Bulman-Fleming and K. McDowell [3] made the observation that a semigroup S is left absolutely flat if and only if S has the free representation extension property. T. E. Hall [5, Remark 11, N. M. Khan (unpublished), and K. Shoji [S] showed that the full transformation semigroups have the representation extension property. Consequently, we have

FINITE BANDS AND AMALGAMATION BASES FOR FINITE SEMIGROUPS

ABSTRACT Hall and Putcha proved that if a finite semigroup S is an amalgamation base for all finite semigroups, then the -classes of S are linearly ordered. Oknin´ski and Putcha proved that any finite semigroup S is an amalgamation base for all finite semigroups if the -classes of are linearly ordered and the semigroup algebra over the complex field has a zero Jacobson radical. In this paper, we study the structure of semigroups which are amalgamation bases for all finite semigroups. In particular, the structure of finite bands which are amalgamation bases for all finite semigroups is determined.

Absolute flatness of regular semigroups with a finite height function

In this paper we give a sufficient condition for regular semigroups with a finite height function to be left absolutely flat. As a consequence, we can show that the semigroup Λ(S) of all right translations of a primitive regular semigroupS with only finitely manyR-classes, with composition being from left to right, is absolutely flat and give a generalization of a Bulman-Fleming and McDowell result concerning absolutely flat semigroups from primitive regular semigroups to regular semigroups with a finite height function. These results give examples of semigroups which are amalgamation bases in the class of semigroups.

The Full Transformation Semigroups of Finite Rank are Amalgamation Bases for Finite Semigroups

In this article, we prove that the full transformation semigroup on a finite set is an amalgamation base for finite semigroups.

Regular Semigroups Which Are Amalgamation Bases for Finite Semigroups

In this paper, we prove that a completely 0-simple (or completely simple) semigroup is an amalgamation base for finite semigroups if and only if it is an amalgamation base for semigroups. By adopting the same method as used in a previous paper, we prove that a finite regular semigroup is an amalgamation base for finite semigroups if its

Decidability of the representation extension property for finite semigroups

{\mathcal J}

Representation extensions and amalgamation bases in rings

-classes are linearly ordered and all of its principal factor semigroups are amalgamation bases for finite semigroups. Finally, we give an example of a finite semigroup U which is an amalgamation base for semigroups, but not all of its principal factor semigroups are amalgamation bases either for semigroups or for finite semigroups.

D*-simple inverse semigroups

We prove that the decision problem of whether or not a finite semigroup has the representation extension property is decidable. 1. The main theorem and preliminaries It is an immediate consequence of the normal form theorem for amalgamated free products of groups that every amalgam of groups embeds in some group. However, this result fails for semigroup amalgams: an early result of Kimura [6] shows that amalgams of semigroups do not necessarily embed in any semigroup (see also [4], Vol. II, page 138). More recently, Sapir [7] has shown that it is in fact undecidable whether an amalgam of (finite) semigroups embeds in any (finite) semigroup. A semigroup S is called an amalgamation base for semigroups if every amalgam of semigroups containing S as a subsemigroup embeds in some semigroup. It is natural to ask if it is decidable whether or not a finite semigroup is an amalgamation base. According to [3], we say that a semigroup S has the representation extension property if for any right S-set XS and any left S-set SM containing S as a left Ssubset, the canonical map: X → X⊗SM (x 7−→ x⊗1) is injective. Hall [5] proved that any semigroup which is an amalgamation base in the class of all semigroups has the representation extension property. In this paper we prove The Main Theorem. It is decidable whether or not a finite semigroup has the representation extension property. Let S be a semigroup. Let M be a nonempty set with a unitary and associative operation of S : S ×M −→ M((s, w) 7−→ sw), where S is the monoid obtained from S by adjoining a new identity 1. Then M is called a left S-set. Dually, a right S-set is defined. If a left S-set [resp. right S-set] M contains elements m1, · · · ,mn such that M = Sm1 ∪ · · · ∪ Smn [resp. M = m1S ∪ · · · ∪mnS], then we say that m1, · · · ,mn are generators of M . A relation ρ on a left [resp. right] S-set M is called an S-congruence if (m,m′) ∈ ρ and s ∈ S implies (sm, sm′) ∈ ρ [resp. (ms,m′s) ∈ ρ]. Let M,N be left [resp. Received by the editors July 1, 1998. 1991 Mathematics Subject Classification. Primary 20M10.

On right self-injective regular semigroups

The main purposes of this paper are to investigate ℤ-injective rings with the representation extension property and its dual, to give a necessary and sufficient condition for a ℤ-injective ring to be an amalgamation base in the class of all rings and to determine structure of ℤ-injective Noetherian rings which are amalgamation bases. Further, in the class of all commutative rings, it is shown that a commutative ring has the representation extension property, if, and only if, it is an amalgamation base.

Regular semigroups with (REP) and (REP) op are not necessarily amalgamation bases

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