On Intra-regular Ordered Semigroups

poe -semigroup -that is an ordered semigroup (:po -semigroup) having a greatest element - is a semilattice of simple semigroups if and only if it is a semilattice of simple poe -semigroups [3].     

On weak commutativity of po-semigroups and their semilattice decompositions

In this paper, concepts of left (right) weakly commutative po-semigroups and r (l or t)-archimedean subsemigroups of a po-semigroup are introduced. Six relations τ, σ, η, ρ, μ, ξ on a po-semigroup are defined. By using them, filters and radicals, fourteen necessary and sufficient conditions in order that a po-semigroup is a semilattice of archimedean subsemigroups are given. The facts that a left weakly commutative (right weakly commutative or weakly commutative) po-semigroup is a semilattice of r (l or t) -archimedean subsemigroups are proved and seven characterizations of these po-semigroups are obtained respectively.     

Ordered semigroups having the <Emphasis Type="Italic">P</Emphasis>-property

In this paper we obtain the following main results. The ordered semigroups which have the P-property are decomposable into archimedean semigroups. Moreover, they are decomposable into semigroupswith the P-property. Conversely, if an ordered semigroup S is a complete semilattice of semigroups which have the P-property, then S itself also has the P-property. An ordered semigroup is CS-indecomposable and has the P-property if and only if it is archimedean. If S is an ordered semigroup, then the relation N:= {(a, b) | N(a) = N(b)} (here N(a) is a filter of S generated by a (a ∈ S)) is the least complete semilattice congruence on S and the class (a) _N is a CS-indecomposable subsemigroup of S for each a ∈ S. We introduce the notion of the P _m -property and describe it in terms of the P-property. Our approach simplifies the proofs of the corresponding results about unordered semigroups. The text was submitted by the authors in English.     

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.     

弱交换富足序半群(Ⅰ)

本文将序半群上的 Ｇｒｅｅｎ’ｓ－关系推广为 Ｇｒｅｅｎ’ｓ＊一关系．给出主序（左、右）＊－理想、主序＊－滤特征描述和弱交换富足序半群的特征．用这些特征证明了一类弱交换富足序半群的结构定理：若序半群Ｓ满足 ,则Ｓ是弱交换富足序半群当且仅当Ｓ是左（右）单序半群｛（ｅ）（Ｓ）｝的半格．     

序半群中的最小完全半素序理想

在序半群上定义了几种新的关系,利用它们得到了序半群上最小完全半素理想的结构,并以此给出了序半群的最小正则半格同余的另一种描述,所得结果是Miroslav Ciric在文[1]中的部分结果向序半群上的推广。     

Dimonoids

It is proved that a system of axioms for a dimonoid is independent and Cayley’s theorem for semigroups has an analog in the class of dimonoids. The least separative congruence is constructed on an arbitrary dimonoid endowed with a commutative operation. It is shown that an appropriate quotient dimonoid is a commutative separative semigroup. The least separative congruence on a free commutative dimonoid is characterized. It is stated that each dimonoid with a commutative operation is a semilattice of Archimedean subdimonoids, each dimonoid with a commutative periodic semigroup is a semilattice of unipotent subdimonoids, and each dimonoid with a commutative operation is a semilattice of a-connected subdimonoids. Various dimonoid constructions are presented.     

Free dimonoids

We characterize the least semilattice congruence of a free dimonoid and prove that a free dimonoid is a semilattice of s-simple subdimonoids each of which is a rectangular band of subdimonoids.     

The Least Regular Order with Respect to a Regular Congruence on Ordered Γ-Semigroups

The motivation mainly comes from the conditions of congruences to be regular that are of importance and interest in ordered semigroups. In 1981, Sen has introduced the concept of the Γ-semigroups. We can see that any semigroup can be considered as a Γ-semigroup. In this paper, we introduce and characterize the concept of the regular congruences on ordered Γ-semigroups and prove the following statements on an ordered Γ-semigroup M : (1) Every ordered semilattice congruences is a regular congruence. (2) There exists the least regular order on the Γ-semigroup M/ρ with respect to a regular congru- ence ρ on M . (3) The regular congruences are not ordered semilattice congruences in general.     

On some congruences of power algebras

In a natural way we can “lift” any operation defined on a set A to an operation on the set of all non-empty subsets of A and obtain from any algebra (A, Ω) its power algebra of subsets. In this paper we investigate extended power algebras (power algebras of non-empty subsets with one additional semilattice operation) of modes (entropic and idempotent algebras). We describe some congruence relations on these algebras such that their quotients are idempotent. Such congruences determine some class of non-trivial subvarieties of the variety of all semilattice ordered modes (modals).     

关于弱交换po－半群

在本文中我们引入弱交换ｐｏ－半群的概念，并研究这类半群到其Ａｒｃｈｉｍｅｄｅｓ子半群的半格分解，给出了这类半群类似于无序半群相应结果的一个刻画。作为推论，我们得到弱交换ｐｏｅ－群和无序半群的相应刻画。     

Certain congruences on E-inversive E-semigroups

A semigroup S is called E-inversive if for every a ∈ S there exists x ∈ S such that ax is idempotent. S is called E-semigroup if the set of idempotents of S forms a subsemigroup. In this paper some special congruences on E-inversive E-semigroups are investigated, such as the least group congruence, a certain semilattice congruence, some regular congruences and a certain idempotent-separating congruence.     

Distributive lattice decompositions of semirings with a semilattice additive reduct

We describe the least distributive lattice congruence on the semirings in the variety of all semirings whose additive reduct is a semilattice, introduce the notion of a k-Archimedean semiring and characterize the semirings that are distributive lattices or chains of k-Archimedean semirings.     

On linearly ordered <Emphasis Type="Italic">H</Emphasis>-closed topological semilattices

We give a criterion for a linearly ordered topological semilattice to be H-closed. We also prove that any linearly ordered H-closed topological semilattice is absolutely H-closed and we show that every linearly ordered semilattice is a dense subsemilattice of an H-closed topological semilattice.     

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.     

交换半群的强半格

本文刻划交换半群的强半格上的最小半格同余，并证明由此得到的商半群为对应的每个交换半群的商半群的强半格。     

Semigroups with the Congruence Extension Property

ρT on a semigroup of T of S extends to the semigroup S, if there exists a congruence _ρ on s such that _{ρ|T= ρT}. A semigroup S has the congruence extension property, CEP, if each congruence on each semigroup extends to S. In this paper we characterize the semigroups with CEP by a set of conditions on their structure (by this we answer a problem put forward in [1]). In particular, every such semigroup is a semilattice of nil extensions of rectangular groups.     

Semilattice modes I: the associated semiring

We examine idempotent, entropic algebras (modes) which have a semilattice term. We are able to show that any variety of semilattice modes has the congruence extension property and is residually small. We refine the proof of residual smallness by showing that any variety of semilattice modes of finite type is residually countable. To each variety of semilattice modes we associate a commutative semiring satisfying 1 +r=1 whose structure determines many of the properties of the variety. This semiring is used to describe subdirectly irreducible members, clones, subvariety lattices, and free spectra of varieties of semilattice modes.Presented by J. Berman.Part of this paper was written while the author was supported by a fellowship from the Alexander von Humboldt Stiftung.     

序半群的滤子与Green关系

给出了半格同余N分别为Green关系L，R，Z和H之一的序半群的刻画。     

Some congruences on trioids

We present the least idempotent congruence on the trioid with a commutative operation, the least semilattice congruence on the trioid with an idempotent operation, and the least separative congruence on the trioid with a commutative operation. Also we construct different examples of trioids.