伪补MS-代数的滤子同余关系

在伪补MS-代数上引入余核滤子和完全滤子的概念,研究伪补MS-代数的余核滤子和完全滤子的性质,获得了余核滤子和完全滤子生成的同余关系的表达式,证明了具有余核滤子的最小同余关系有同余一致性.     

Semigroups whose lattices of right congruences are semiatomic

The main theorems presented here are characterizations of a semigroup with a left identity whose lattice of right congruences is semiatomic. These theorems are preceded by a number of results on minimal right congruences.     

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).     

On right congruences of semigroups having no proper essential right congruences

A right congruence ?? in a semigroup S is essential if for any right congruence ?? we have ??????=?? (the identity relation) implies ??=??. Clearly, the universal relation, ??, is an essential right congruence. We say ?? is proper if ??????. In this paper we get a necessary and sufficient condition for a semigroup with an identity element?1 and having no proper essential right congruences to have a distributive lattice of right congruences.     

关于左型A半群上的fuzzy同余

引入了左富足半群上fuzzy右好同余和fuzzy右消去同余的概念,给出了左富足半群上fuzzy右好同余的性质和特征.在此基础上,给出了左型A半群上fuzzy右好同余和fuzzy右消去同余的性质.得到了左型A半群上的fuzzy右好同余为fuzzy右消去同余的充要条件.     

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.     

The order of principal congruences of a bounded lattice

We characterize the order of principal congruences of a bounded lattice (also of a complete lattice and of a lattice of length 5) as a bounded ordered set. We also state a number of open problems in this new field.     

Lattice congruences, fans and Hopf algebras

We give a unified explanation of the geometric and algebraic properties of two well-known maps, one from permutations to triangulations, and another from permutations to subsets. Furthermore we give a broad generalization of the maps. Specifically, for any lattice congruence of the weak order on a Coxeter group we construct a complete fan of convex cones with strong properties relative to the corresponding lattice quotient of the weak order. We show that if a family of lattice congruences on the symmetric groups satisfies certain compatibility conditions then the family defines a sub Hopf algebra of the Malvenuto–Reutenauer Hopf algebra of permutations. Such a sub Hopf algebra has a basis which is described by a type of pattern avoidance. Applying these results, we build the Malvenuto–Reutenauer algebra as the limit of an infinite sequence of smaller algebras, where the second algebra in the sequence is the Hopf algebra of non-commutative symmetric functions. We also associate both a fan and a Hopf algebra to a set of permutations which appears to be equinumerous with the Baxter permutations.     

Extension of congruences on semirings of continuous functions

The lattices of congruences of semirings and semifields of continuous nonnegative functions over an arbitrary topological space are studied. It is proved that congruences of the semifield of continuous positive functions can be extended to congruences of the semiring of continuous nonnegative functions.     

含幺元的半格和群的Zappa-Sz\'{e}p积上的同余

Let P = E G be a Zappa-Szp product of a semilattice E with an identity and a group G. In this paper, we first introduce the concept of congruence pairs for P , and then prove that every congruence on P can be described by such a congruence pair. In fact the congruence lattice on P is lattice-isomorphic to the set of all congruence pairs for P . Finally,we characterize group congruences on P .     

On the representation of distributive semilattices

This paper gives new sufficient conditions for the representability of a distributive semilattice as the semilattice of compact congruences of a lattice. The main theorem includes all previously known results of this type.Dedicated to Bjarni Jónsson on his 70th birthday     

The Lattice of Fully Invariant Subgroups of the Cotorsion Hull

In this paper, we consider the lattice of fully invariant subgroups of the cotorsion hull of a separable p-group, which is assumed to be an arbitrary direct sum of torsion-complete p-groups. It is shown that this lattice is isomorphic to the lattice of filters of a lower semilattice made up of infinite matrices.     

含幺逆半群的半格上的同余格

本文证明了含幺逆半群Sα（α∈Y）的半格的容许同余格同构于半格S上同余格的一个子格。     

Congruences on near-Heyting algebras

A near-Heyting algebra is a join-semilattice with a top element such that every principal upset is a Heyting algebra. We establish a one-to-one correspondence between the lattices of filters and congruences of a near-Heyting algebra. To attain this aim, we first show an embedding from the lattice of filters to the lattice of congruences of a distributive nearlattice. Then, we describe the subdirectly irreducible and simple near-Heyting algebras. Finally, we fully characterize the principal congruences of distributive nearlattices and near-Heyting algebras. We conclude that the varieties of distributive nearlattices and near-Heyting algebras have equationally definable principal congruences.     

幂等元位于中心的半群的局部化和最小幂幺半群同余

局部化是交换代数的重要工具[1],证明幂等元位于中心的半群在其幂等元半格上的局部化存在且唯一,并给出此类半群的最小幂幺半群同余.另外,给出了若干半群的重要同余的刻划.     

On a class of Ribaucour’s normal congruences

The purpose of this paper is to determine all of Ribaucour’s normal congruences such that the invariants of the Laplace equation, corresponding to the conjugate double system of curves of the images of the developables of the sphere, vanish. We give a new proof of the existence ofI-congruences and make some observations concerning Bianchi’s conjecture relative to the congruences of Thybaut. To the memory of Alessandro Terracini     

Block-separating Congruence Lattices of E-bisimple Semigroups

In this paper,we study block-separating congruences on E-bisimple semigroups. We describe two congruence relations T and W on the lattice of block-separating congruences, where the latter is defined in this paper. We determine the greatest and smallest congruences in T-classes and W-classes.     

Line congruences as surfaces in the space of lines

The space of lines in R³ can be viewed as a four dimensional homogeneous space of the group of Euclidean motions, E(3). Line congruences arise in the classical method of transforming one surface to another by lines. These transformations are particularly interesting if some geometric property of the original surface is preserved. Line congruences, then, are two parameter families of lines and can be studied as surfaces in the space of lines. In this paper, we use the method of moving frames to study line congruences. We calculate the first order invariants of line congruences for which there are two real focal surfaces, and give the geometric meaning of these invariants. We look specifically at the case where the two first order invariants are constant and give a simple proof of Bäcklund's Theorem which relates to the transformation of one pseudospherical surface, a surface of constant negative Gaussian curvature, to another. These transformations are of interest since pseudospherical surfaces correspond to solutions to the sine-Gordon equation. We also give a proof of Bianchi's permutability theorem for pseudospherical surfaces in this context. Finally, we use the results of these theorems to generate some pseudospherical surfaces. All of these concepts and results are understood in terms of the structure equations of the line congruence.     

The direct product of right congruences

In a number of earlier papers the study of the structure of semigroups has been approached by means of right congruences. Such an approach seems appropriate since a right congruence is one of the possible analogs of both the right ideal of a ring and the subgroup in a group. Each of these substructures plays a strong role in the study of the structure of their respective systems. In both the ring and the group the internal direct product is naturally, and effectively, defined. However, what such an internal direct product should be for two right congruences of a semigroup is not so clear. In this paper we will offer a possible definition and consider some of the consequences of it. We will also extend some of these results to automata.     

Coupled Right Orthosemirings Induced by Orthomodular Lattices

L. P. Belluce, A. Di Nola and B. Gerla established a connection between MV-algebras and (dually) lattice ordered semirings by means of so-called coupled semirings. A similar connection was found for basic algebras and semilattice ordered right near semirings by the authors. The aim of this paper is to derive an analogous connection for orthomodular lattices and certain semilattice ordered near semirings via so-called coupled right orthosemirings.