Semilattice decompositions of semigroups

The purpose of this paper is to develop a general theory of semilattice decompositions of semigroups from the point of view of obtaining theorems of the type: A semigroup S has propertyD if and only if S is a semilattice of semigroups having property β. As such we are able to extend the theories of Clifford [3], Andersen [1], Croisot [5], Tamura and Kimura [14], Petrich [9], Chrislock [2], Tamura and Shafer [15], Iyengar [7] and Weissglass and the author [10]. The root of our whole theory is Tamura's semilattice decomposition theorem [12, 13]. Of this, we give a new proof. The results of this paper were obtained by the author between January and July of 1971, while an undergraduate at the University of California, Santa Barbara.     

OnP-Q ordered semigroups

In this paper,P-ordered andQ-ordered semigroups are studied. Some characterizations and properties of such semigroups are obtalned. Also the relationship between maximal (minimum) regular ordered semigroups and unitary regular semigroups is investigated.Research is partiallysupported by CUHK grant No. 220.600.080.     

CS-indecomposable ordered semigroups

An ordered semigroup S is called CS-indecomposable if the set S × S is the only complete semilattice congruence on S. In the present paper we prove that each ordered semigroup is, uniquely, a complete semilattice of CS-indecomposable semigroups, which means that it can be decomposed into CS-indecomposable components in a unique way. Furthermore, the CS-indecomposable ordered semigroups are exactly the ordered semigroups that do not contain proper filters. Bibliography: 6 titles. Published in Zapiski Nauchnykh Seminarov POMI, Vol. 343, 2007, pp. 222–232.     

Extensions of ordered semigroups

The present note summarizes the author's dissertation [2]. Let S and T be ordered semigroups, the latter with a zero element O. Let Σ be an ideal extension of S by T determined by a partial homomorphism ϕ of T^*=T/O into S. A partial solution is given to the problem of determining all ways of extending the given orders on S and T^* to Σ=S U T^*. Every such “extending order” (≤) on Σ carries with it a certain “null decomposition” T^*=X∪Y (X∩Y=ℴ) of T^*, and the existence and behavior of extending orders is discussed in terms of properties of null decompositions of T^*.     

On intra-regular ordered semigroups

Communicated by J. S. Ponizovski?     

Naturally ordered transformation semigroups

The full transformation semigroup (T _x o) on the setX is studied with respect to the natural partial order which is defined on any regular semigruop. A characterization of this ordering in terms of images and kernels is given, the maximal and minimal elements are determined and the covering elements are described. Lower and upper bounds for two transformations are studied and necessary as well as sufficient conditions for their existence are gieven.     

Pseudoorder in ordered semigroups

Communicated by J. S. Ponizovskiį     

Subdirect decompositions of finitely generated commutative semigroups

All finitely generated commutative semigroups which do not have proper finite subdirect decompositions are determined. This yields subdirect decompositions of finitely generated commutative semigroups and some idea of their structure.     

Wreath products of ordered semigroups

In this paper ordered wreath products of ordered monoids by ordered acts are investigated. In 4. we characterize idempotent isotone wreath products. In 3. the monoid of order preserving endomorphisms of a free ordered act is represented as Cartesian ordered isotone wreath product. Moreover, we give conditions for this wreath product to be I-regular.     

Semilattice composition of ordered semigroups

We study the semilattice composition of ordered semigroups (a concept opposite to that of the semilattice decomposition), using the ideal extensions. The text was submitted by the authors in English.     

Wreath products of ordered semigroups

In this part of the paper we give necessary and sufficient conditions for ordered wreath products of ordered semigroups by ordered acts to be inverse. In addition for Cartesian ordered wreath products we give conditions under which passing to inverses is isotone or antitone.     

The chain of right simple semigroups in ordered semigroups

The relation ≤ is defined on the set of right ideals of an ordered semigroup. The main result of this paper is as follows: an ordered semigroup S is a chain of right simple ordered semigroups if and only if ≤ is an order relation. Bibliography: 3 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 227, 1995, pp. 83–88.