Maximal and Minimal Semilattices on Ordered Sets

The map which takes an element of an ordered set to its principal ideal is a natural embedding of that ordered set into its powerset, a semilattice. If attention is restricted to all finite intersections of the principal ideals of the original ordered set, then an embedding into a much smaller semilattice is obtained. In this paper the question is answered of when this construction is, in a certain arrow-theoretic sense, minimal. Specifically, a characterisation is given, in terms of ideals and filters, of those ordered sets which admit a so-called minimal embedding into a semilattice. Similarly, a candidate maximal semilattice on an ordered set can be constructed from the principal filters of its elements. A characterisation of those ordered sets that extend to a maximal semilattice is given. Finally, the notion of a free semilattice on an ordered set is given, and it is shown that the candidate maximal semilattice in the embedding-theoretic sense is the free object.     

Idempotent distributive semirings II

In [2] it is shown that every idempotent distributive semiring is the P?onka sum of a semilattice ordered system of idempotent distributive semirings which satisfy the generalized absorption law x+xyx+x=x. We shall show that an idempotent distributive semiring which satisfies the above absorption law must be a subdirect product of a distributive lattice and a semiring which satisfies the additional identity xyx+x+xyx=xyx. Using this, we construct the lattice of all equational classes of idempotent distributive semirings for which the two reducts are normal bands.     

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 the additive structure of a class of ordered semirings

In this paper we shall determine the additive structure of totally ordered semirings if its multiplicative structure is a right naturally ordered o-Archimedean semigroup.     

On Ordered Semigroups Which are Semilattices of Simple and Regular Semigroups

We characterize the ordered semigroups which are decomposable into simple and regular components. We prove that each ordered semigroup which is both regular and intra-regular is decomposable into simple and regular semigroups, and the converse statement also holds. We also prove that an ordered semigroup S is both regular and intra-regular if and only if every bi-ideal of S is an intra-regular (resp. semisimple) subsemigroup of S. An ordered semigroup S is both regular and intra-regular if and only if the left (resp. right) ideals of S are right (resp. left) quasi-regular subsemigroups of S. We characterize the chains of simple and regular semigroups, and we prove that S is a complete semilattice of simple and regular semigroups if and only if S is a semilattice of simple and regular semigroups. While a semigroup which is both π-regular and intra-regular is a semilattice of simple and regular semigroups, this does not hold in ordered semigroups, in general.     

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 semiring of 1-preserving endomorphisms of a semilattice

We prove that the semirings of 1-preserving and of 0,1-preserving endomorphisms of a semilattice are always subdirectly irreducible and we investigate under which conditions they are simple. Subsemirings are also investigated in a similar way.     

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.     

Examples concerning absolute flatness and amalgamation in pomonoids

Flatness properties of acts over monoids have been studied for almost four decades and a substantial literature is now available on the subject. Analogous research dealing with partially ordered monoids acting on posets was begun in the 1980s in two papers by S.M. Fakhruddin, and, after a dormancy period of some 20 years, has recently been rekindled with the appearance of several research articles. In comparing flatness properties of S-acts and S-posets, it has been noted that the imposition of order results in severe restrictions as far as absolute flatness is concerned. For example, whereas every inverse monoid is absolutely flat (meaning all of its left and right acts are flat), even the three-element chain in its natural order, considered as a pomonoid, fails to have this property. It has long been understood that absolutely flat monoids, in particular, inverse monoids, are amalgamation bases in the class of all monoids. The purpose of the present article is to further investigate absolute flatness of pomonoids and to begin to study its connection with amalgamation in that context. T.E. Hall’s results, that amalgamation bases in the class of all monoids have the so-called representation extension property (REP), which in turn implies the right congruence extension property, are first adapted to the ordered context. A detailed study of the compatible orders (of which there are exactly 13) on the three-element chain semilattice U then reveals a wide range of possibilities: exactly four of these orders render U absolutely flat as a pomonoid, two more give it the right order-congruence extension property in every extension (RCEP) (but fail to make it an amalgamation base because of the failure of the ordered analogue of (REP)), and for the remaining seven, even (RCEP) fails.     

阿基米德序半群的链的若干刻画

本文首先引入了一个序半群$S$的准素模糊理想的概念,通过序半群$S$上的一些二元关系以及它的理想的模糊根给出了该序半群是阿基米德序子半群的半格的一些刻画.进一步地借助于序半群$S$的模糊子集对该序半群是阿基米德序子半群的半格进行了刻画.尤其是通过序半群的模糊素根定理证明了序半群$S$是阿基米德序子半群的链当且仅当$S$是阿基米德序子半群的半格且$S$的所有弱完全素模糊理想关于模糊集的包含关系构成链.     

On the Least Property of the Semilattice Congruences on PO-Semigroups

n on po-semigroups. We study the least property of (ordered) semilattice congruences, and prove: 1. N is the least ordered semilattice congruence on pr-semigroups (cf.[1]). 2. n is the least semilattice congruence on po-semigroups. 3. N is not the least semilattice congruence on po-semigroups in general. Thus, we give a complete solution to the problem posed by N. Kehayopulu in [1].     

偏序半群的最大自然序半格同态象

引进了自然序半格，偏序半群的自然序半格同态象和二次主根基等概念，利用二次主根基构造出了任意偏序半群的最大自然序半格同态象。     

几类加法非正则半环

In this paper, we introduce Green＇s .-relations on semirings and define [left, right] adequate semirings to explore additively non-regular semirings. We characterize the semirings which are strong b-lattices of [left, right] skew-halfrings. Also, as further generalization, the semirings are described which are subdirect products of an additively commutative idempotent semiring and a [left, right] skew-halfring. We extend results of constructions of generalized Clifford semirings （given by M. K. Sen, S. K. MaRy, K. P. Shum, 2005） and the semirings which are subdirect products of a distributive lattice and a ring （given by S. Ghosh, 1999） to additively non-regular semirings.     

The Semigroup Structure of Left Clifford Semirings

In this paper,we generalize Clifford semirings to left Clifford semirings by means of the so-called band semirings.We also discuss a special case of this kind of semirings,that is, strong distributive lattices of left rings.     

乘法半群为左正规纯正群的半环

研究了加法半群为半格,乘法半群为左正规纯正群的半环．证明了此类半环（S,＋,．）可以嵌入到半格（S,＋）的自同态半环中．构造S的一个特定的偏序关系,得到了（S,·）上的自然偏序与所构造偏序相等的等价条件．     

Varieties Generated by Ordered Bands I

Ordered bands are regarded as semirings whose multiplicative reduct is a band and whose additive reduct is a chain. We find the variety of semirings generated by all ordered bands and we determine part of the lattice of its subvarieties.     

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.     

FP-injective semirings,semigroup rings and Leavitt path algebras

In this paper we give characterisations of FP-injective semirings (previously termed “exact” semirings in work of the first author). We provide a basic connection between FP-injective semirings and P-injective semirings, and establish that FP-injectivity of semirings is a Morita invariant property. We show that the analogue of the Faith-Menal conjecture (relating FP-injectivity and self-injectivity for rings satisfying certain chain conditions) does not hold for semirings. We prove that the semigroup ring of a locally finite inverse monoid over an FP-injective ring is FP-injective and give a criterion for the Leavitt path algebra of a finite graph to be FP-injective.     

Convolution algebras over complete semirings

Constructions of complete semirings via categorical algebra are provided, that is, we construct convolution algebras of given complete semirings via so-called action networks (i.e., small covariant categories). We aim to get some structural insight into the principle of unrestricted convolution.