纯正半群上的同余扩张(一)

刻划半群上的同余及其扩张是半群的代数理论中的一个非常重要的课题．本文讨论了带上的同余的正规性和不变性以及在其Ｈａｌｌ半群上的扩张，从同余扩张的角度刻划了带上的同余的性质，给出了扩张的极大、极小同余的描述．     

A class of formal languages

We deal with languages that are classes of fully invariant congruences on free semigroups of finite rank. The question is posed as to whether a given fully invariant congruence coincides with a syntactic congruence of the language in question. If all classes of a given fully invariant congruence are rational languages, the corresponding variety is then said to be rational. A number of properties of rational varieties is established—in particular, we point to the way in which they are related to finite varieties. Translated fromAlgebra i Logika, Vol. 37, No. 4, pp. 478–492, July–August, 1998.     

Algebraic Characterizations for Universal Fragments of Logic

In this paper we address our efforts to extend the well-known connection in equational logic between equational theories and fully invariant congruences to other–possibly infinitary–logics. In the special case of algebras, this problem has been formerly treated by H. J. Hoehnke [10] and R. W. Quackenbush [14]. Here we show that the connection extends at least up to the universal fragment of logic. Namely, we establish that the concept of (infinitary) universal theory matches the abstract notion of fully invariant system. We also prove that, inside this wide group of theories, the ones which are strict universal Horn correspond to fully invariant closure systems, whereas those which are universal atomic can be characterized as principal fully invariant systems.     

Relatively free bands

The operators h_n and i _n and their duals hmacr;_n and imacr;_n defined on the free semigroup X ⁺ for a nonempty set X by Gerhard and Petrich are proved here to be homomorphisms of onto certain subsets of X ⁺ with a suitable multiplication. From the work of the authors mentioned, these operators induce fully invariant congruences on X ⁺ corresponding to join irreducible varieties of bands if X is countably infinite. New operators on X ⁺ are defined by means of these operators which give homomorphisms in an analogous way and induce fully invariant congruences on X ⁺corresponding to all varieties of bands except for the variety of all bands, and some varieties of normal bands. The former of these was investigated by the mentioned authors and the latter must be treated differently. By means of the above operators we are able to characterize all cases of relatively free bands.     

E-varieties of regular semigroups,relatively bifree objects and fully invariant congruences

Analogous to the concept of a free object on a setX in a variety of algebras is the notion of a bifree object onX in an e-variety of regular semigroups. If an e-variety contains a bifree object onX, then a homomorphic image of that bifree object is itself bifree onX in some e-variety if and only if the corresponding congruence is fully invariant. Furthermore, the lattice of e-subvarieties of any locally inverse or E-solid e-variety ε is antiisomorphic with the lattice of all fully invariant congruences on the bifree object on a countably infinite setX in ε. We give a Birkhoff-type theorem for classes of locally inverse or E-solid semigroups, and we give an intrinsic test for whether or not a regular semigroup is bifree onX in the e-variety it generates.     

Fully invariant and verbal congruence relations

A congruence relation θ on an algebra A is fully invariant if every endomorphism of A preserves θ. A congruence θ is verbal if there exists a variety ${\mathcal{V}}$ such that θ is the least congruence of A such that ${{\bf A}/\theta \in \mathcal{V}}$ . Every verbal congruence relation is known to be fully invariant. This paper investigates fully invariant congruence relations that are verbal, algebras whose fully invariant congruences are verbal, and varieties for which every fully invariant congruence in every algebra in the variety is verbal.     

Semigroup Varieties on Whose Free Objects Almost All Fully Invariant Congruences are Weakly Permutable

A semigroup variety is said to be of index 2 if all nil-semigroups of the variety are semigroups with zero multiplication. We describe all semigroup varieties V of index 2 on free objects of which every two fully invariant congruences contained in the least semilattice congruence are weakly permutable, and semigroup varieties of index 2 all of whose subvarieties share the above-mentioned property.     

Congruences on *-Regular Semigroups

In this paper we study the congruences of *-regular semigroups, involution semigroups in which every element is p-related to a projection (an idempotent fixed by the involution). The class of *-regular semigroups was introduced by Drazin in 1979, as the involutorial counterpart of regular semigroups. In the standard approach to *-regular semigroup congruences, one ,starts with idempotents, i.e. with traces and kernels in the underlying regular semigroup, builds congruences of that semigroup, and filters those congruences which preserve the involution. Our approach, however, is more evenhanded with respect to the fundamental operations of *-regular semigroups. We show that idempotents can be replaced by projections when one passes from regular to *-regular semigroup congruences. Following the trace-kernel balanced view of Pastijn and Petrich, we prove that an appropriate equivalence on the set of projections (the *-trace) and the set of all elements equivalent to projections (the *-kernel) fully suffice to reconstruct an (involution-preserving) congruence of a *-regular semigroup. Also, we obtain some conclusions about the lattice of congruences of a *-regular semigroup. This revised version was published online in June 2006 with corrections to the Cover Date.     

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.     

Curvature properties of 6-parametric robot manipulators

The Lie group C₆ of all orientation preserving congruences of the Euclidean space E₃ has a natural invariant pseudo-Riemannian structure determined by the Klein quadratical form of its Lie algebra.In the paper we connect the acceleration properties of a 6-parametric robot manipulator with the properties of the Levi-Civita connection and its curvature tensor of the pseudo-Riemannian structure of C₆.The paper is a continuation of [3].     

Intermediately fully invariant subgroups of abelian groups

Describing intermediately fully invariant subgroups of divisible and torsion groups, we show that the intermediately fully invariant subgroups are direct summands in a completely decomposable group whose every homogeneous component is decomposable. For torsion groups, we find out when all their fully invariant subgroups are intermediately fully invariant; and for torsion-free groups, this question comes down to the reduced case. Also, in a torsion group that is the sum of cyclic subgroups, its subgroup is shown to be intermediately inert if and only if it is commensurable with some intermediately fully invariant subgroup.     

Semigroup Varieties with 1.5-Permutable Fully Invariant Congruences on their Free Objects

We describe semigroup varieties on whose free objects the product of any two fully invariant congruences coincides with their set-theoretical union. The work was supported by the Russian Foundation of Basic Research (Grant 01-01-00258) and by the program Russian Universities – Basic Research of the Ministry of Education of the Russian Federation (project No. 04.01.59).Mathematics Subject Classifications (2000) 20M07, 20M05, 08B15.     

Congruences between Siegel modular forms

Using the moduli theory of abelian varieties and a recent result of Böcherer-Nagaoka on lifting of the generalized Hasse invariant, we show congruences between the weights of Siegel modular forms with congruent Fourier expansions. This result implies that the weights of p-adic Siegel modular forms are well defined.     

序半群的一类正则同余

设S是有向序半群,本文给出了S上的一类正则同余,称为强序同余的定义及性质.证明了S的强序同余是强正则同余,但反之不成立.同时证明了强序同余格SOC(S)是S的同余格C(S)关于通常集合的交和传递积的V-完备的分配子格.     

On congruences in closed post classes

It is shown herein that on almost all closed Post classes only trivial congruences are possible. Closed Post classes having nontrivial congruences are indicated, and all congruences of these classes are described. For k3 closed classes having an infinity of congruences are indicated.Translated from Matematicheskie Zametki, Vol. 13, No. 5, pp. 725–734, May, 1973.The author is grateful to S. V. Yablonskii for valuable comments and assistance in the research.     

Separable and vector groups whose projectively invariant subgroups are fully invariant

Siberian Mathematical Journal - We study the abelian groups all of whose projectively invariant subgroups are fully invariant. We describe the separable and vector groups with this property.     

Congruence computations in principal arithmetical varieties

This paper is a continuation of the earlier paper by the same authors in which a primary result was that every arithmetical affine complete variety of finite type is a principal arithmetical variety with respect to an appropriately chosen Pixley term. The paper begins by presenting an extension of this result to all finitely generated congruences and, as an example, constructs a closed form solution formula for any finitely presented system of pairwise compatible congruences (the Chinese remainder theorem). It is also shown that in all such varieties the meet of principal congruences is also principal, and finally, if a minimal generating algebra of the variety is regular, it is shown that the variety is also regular and the join of principal congruences is again principal.     

分配半环上的可除半环同余

§ 1.　Introduction　　AsemiringSisanalgebraicsystem (S ,+ ,·)consistingofanon_emptysetStogetherwithtwobinaryoperations +and·onSsuchthat (S ,+ )and (S ,·)aresemigrolupscon nectedbyring_likedistributivity .AsemiringSiscalleddistributiveifinStheadditionisdis tributiveaboutmultiplication ,i.e .ab+c=(a+c) (b +c)anda+bc=(a +b) (a+c)holdforalla ,b ,c∈S .AsemiringSiscalleddivisibleif(S ,·)isagroup .AnequivalencerelationρonasemiringSiscalledacongruenceonS ,iffρisacongruenceon (S ,+ )and (…     

Realization of all Dold's congruences with stability

The main goal of this paper is to prove that for each n>2, every sequence of integers satisfying Dold's congruences is realized as the sequence of fixed point indices of the iterates of an orientation preserving Rn-homeomorphism at an isolated stable fixed point. We use Conley index techniques even though stable fixed points are not isolated invariant sets.