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.     

Chain decompositions of ordered semigroups

Characterizations of ordered semigroups which can be decomposed into (natural ordered) chains of ω -simple ordered semigroups are given, where ω -simple ordered semigroups are ξ _l (ξ _t ) -simple, left (t -) simple, L _n (H _n ) -simple, l (t )-archimedean and nil-extensions of left (t -) simple ordered semigroups, respectively. As a generalization of the theory of Clifford semigroups (without orders) to ordered semigroups, ordered semigroups which are semilattices of t -simple subsemigroups are characterized.     

Intuitionistic fuzzy quasi-ideals of ordered semigroups

In this paper we define intuitionistic fuzzy quasi-ideals of ordered semigroups. The main result of the paper is a characterization of quasi-ideals in terms of intuitionistic fuzzy quasi-ideals. We also characterize left simple, right simple, and completely regular ordered semigroups in terms of intuitionistic fuzzy quasi-ideals. We study the decomposition of left and right simple ordered semigroups using intuitionistic fuzzy quasi-ideals.     

单的非负有序关联半群

一个负有序关联半群(S,≤,.,*)称为单的关联半群,如果S的所有滤子是{1}和S本身.对负有序关联半群是单的关联半群进行了刻画,给出了一个负的关联半群是单的关联半群的等价条件.     

Ordered semigroups whose elements are separated by prime ideals

In this paper we add one more characterization of intra-regular ordered semigroups in the existing bibliography by proving that the ordered semigroups whose elements are separated by prime ideals are actually the intra-regular ordered semigroups, a result which generalizes the corresponding result of semigroups (without order).     

Homomorphisms of implicative semigroups

Implicative semigroups and Brouwerian semigroups were studied by W. C. Nemitz and T. S. Blyth, respectively. In this paper, following the ideas of Nemitz and Blyth, we introduce the notion of negatively partially ordered implicative semigroups and studied the homomorphisms between these semigroups. Some results of Nemitz on implicative semilattices are generalized and amplified to implicative semigroups. Research is partially supported by CUHK small Project grant 220.600.080.     

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.     

Precoherent quantale completions of partially ordered semigroups

Just as complete lattices can be viewed as the completions of posets, quantales can also be treated as the completions of partially ordered semigroups. Motivated by the study on the well-known Frink completions of posets, it is natural to consider the “Frink” completions for the case of partially ordered semigroups. For this purpose, we firstly introduce the notion of precoherent quantale completions of partially ordered semigroups, and construct the concrete forms of three types of precoherent quantale completions of a partially ordered semigroup. Moreover, we obtain a sufficient and necessary condition of the Frink completion on a partially ordered semigroup being a precoherent quantale completion. Finally, we investigate the injectivity in the category $$\mathbf {APoSgr}_{\le }$$ of algebraic partially ordered semigroups and their submultiplicative directed-supremum-preserving maps, and show that the $$\mathscr {E}_{\le }$$-injective objects of algebraic partially ordered semigroups are precisely the precoherent quantales, here $$\mathscr {E}_{\le }$$ denote the class of morphisms $$h:A\longrightarrow B$$ that preserve the compact elements and satisfy that $$h(a_1)\cdots h(a_n)\le h(b)$$ always implies $$a_1\cdots a_n\le b$$.     

Compact operator semigroups applied to dynamical systems

A right chain ordered semigroup is an ordered semigroup whose right ideals form a chain. In this paper we study the ideal theory of right chain ordered semigroups in terms of prime ideals, completely prime ideals and prime segments, extending to these semigroups results on right chain semigroups proved in Ferrero et al. (J Algebra 292:574–584, 2005).     

Positive one-parameter semigroups on ordered banach spaces

In this review we describe the basic structure of positive continuous one-parameter semigroups acting on ordered Banach spaces. The review is in two parts.First we discuss the general structure of ordered Banach spaces and their ordered duals. We examine normality and generation properties of the cones of positive elements with particular emphasis on monotone properties of the norm. The special cases of Banach lattices, order-unit spaces, and base-norm spaces, are also examined.Second we develop the theory of positive strongly continuous semigroups on ordered Banach spaces, and positive weak^*-continuous semigroups on the dual spaces. Initially we derive analogues of the Feller-Miyadera-Phillips and Hille-Yosida theorems on generation of positive semigroups. Subsequently we analyse strict positivity, irreducibility, and spectral properties, in parallel with the Perron-Frobenius theory of positive matrices.     

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.     

Cone-transitive matrix semigroups

Semigroups of matrices (over an ordered field) with non-negative entries are considered. A complete characterization is obtained for the semigroups which are minimal transitive on the positive (or non-negative) cone of the underlying vector space. Consequently, an explicit form for the semigroups sharply transitive on the cone is derived.     

Cone-transitive matrix semigroups

Semigroups of matrices (over an ordered field) with non-negative entries are considered. A complete characterization is obtained for the semigroups which are minimal transitive on the positive (or non-negative) cone of the underlying vector space. Consequently, an explicit form for the semigroups sharply transitive on the cone is derived.     

Characterization of non-nilpotent topological interval semigroups

A remarkable subclass of linearly ordered semigroups, called interval semigroups, defined on connected and compact sets is studied. Particularly, a generalized notion of o-isomorphism, called weak o-embedding, of such semigroups into the real numbers with standard operations is given. A representation theorem for the weak o-embedding of topological Archimedean interval semigroups with no zero divisors is provided. Such characterization is shown to be the best one possible.     

Undecidability of representability for lattice-ordered semigroups and ordered complemented semigroups

We prove that the problems of representing a finite ordered complemented semigroup or finite lattice-ordered semigroup as an algebra of binary relations over a finite set are undecidable. In the case that complementation is taken with respect to a universal relation, this result can be extended to infinite representations of ordered complemented semigroups.     

On left regular and intra-regular ordered semigroups

We study the decomposition of left regular ordered semigroups into left regular components and the decomposition of intra-regular ordered semigroups into simple or intra-regular components, adding some additional information to the results considered in [KEHAYOPULU, N.: On left regular ordered semigroups, Math. Japon. 35 (1990), 1057–1060] and [KEHAYOPULU, N.: On intra-regular ordered semigroups, Semigroup Forum 46 (1993), 271–278]. We prove that an ordered semigroup S is left regular if and only if it is a semilattice (or a complete semilattice) of left regular semigroups, equivalently, it is a union of left regular subsemigroups of S. Moreover, S is left regular if and only if it is a union of pairwise disjoint left regular subsemigroups of S. The right analog also holds. The same result is true if we replace the words “left regular” by “intraregular”. Moreover, an ordered semigroup is intra-regular if and only if it is a semilattice (or a complete semilattice) of simple semigroups. On the other hand, if an ordered semigroup is a semilattice (or a complete semilattice) of left simple semigroups, then it is left regular, but the converse statement does not hold in general. Illustrative examples are given.     

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.     

Archimedeaness and additive utility on totally ordered semigroups

We study problems concerning the existence of additive utility funtions defined on totally ordered semigroups. The existence of an additive utility function on a semigroup is characterized by means of conditions that are similar, but not equivalent, to Archimedeaness. This fact is used to analyze the existence of utility representations (not necessarily additive) on totally ordered Abelian groups. In this direction, we show that the positive cone of a representable totally ordered Abelian group admits a countable partition into Archimedean semigroups. All the semigroups in that partition are representable by means of a utility function, but at most one is additively representable. Communicated by M. W. Mislove     

Soft ordered semigroups

Molodtsov introduced 1999 the concept of soft set as a new mathematical tool for dealing with uncertainties that is free from the difficulties that have troubled the usual theoretical approaches. In this paper we apply the notion of soft sets by Molodtsov to ordered semigroups. The notions of (trivial, whole) soft ordered semigroup, soft ordered subsemigroup, soft left (right) ideal, and left (right) idealistic soft ordered semigroup are introduced, and various related properties are investigated (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.