Orthodox Congruences on An Eventually Regular Semigroup

It is shown that each regular congruence on an eventually regular semigroup is uniquely determined by its kernel and hyper-trace. Furthermore, the orthodox congruences (resp., the regular congruences ) on an eventually regular (resp., orthodox) semigroup S are described by means of certain congruence pair (ξ, K), where ξ is a certain normal congruence on the subsemigroup 〈E(S)〉 generated by E(S) and K is a certain normal subsemigroup of S.     

Partial kernel normal systems in regular semigroups

\noindent Let P be an arbitrary full subset of the set E(S) of the idempotents in a regular semigroup S . We shall prove that each congruence on S is completely determined by its partial kernel normal system linking to P and give the kernel normal systems in regular semigroups an abstract characterization. January 19, 1998     

纯正半群上的强同余及Clifford同余

把Reilly对逆半群的幂等元集合的正规划分的概念推广到纯正半群,用它从另一角度刻画。了纯正半群上强同余的结构.并刻画了具有T关系的两个强同余的联和交的正则核正规系,又讨论了纯正半群上的Clifford同余,给出了最小Clifford同余的刻画.     

Congruences on regular semigroups

The kernel of a congruence on a regular semigroup S may be characterized as a set of subsets of S which satisfy the Teissier-Vagner-Preston conditions. A simple construction of the unique congruence associated with such a set is obtained. A more useful characterization of the kernel of a congruence on an orthodox semigroup (a regular semigroup whose idempotents form a subsemigroup) is provided, and the minimal group congruence on an orthodox semigroup is determined.     

Monotone normality and neighborhood assignments

We prove that every neighborhood assignment for a monotonically normal space has a kernel which is homeomorphic to some subspace of an ordinal. As a corollary, every monotone neighborhood assignment for a monotonically normal space has a discrete kernel, which gives a partial answer to a question posed in Buzyakova et al. (2007) [5]. We also give an example of a regular space which has a neighborhood assignment with no kernels homeomorphic to any subspace of an ordinal.     

Sublattices of the Lattices of Strong P-Congruences on P-Inversive Semigroups

In this paper, we describe strong P-congruences and sublattice-structure of the strong P-congruence lattice C_P(S) of a P-inversive semigroup S(P). It is proved that the set of all strong P-congruences C_P(S) on S(P) is a complete lattice. A close link is discovered between the class of P-inversive semigroups and the well-known class of regular ⋆-semigroups. Further, we introduce concepts of strong normal partition/equivalence, C-trace/kernel and discuss some sublattices of C_P(S). It is proved that the set of strong P-congruences, which have C-traces (C-kernels) equal to a given strong normal equivalence of P (C-kernel), is a complete sublattice of C_P(S). It is also proved that the sublattices determined by C-trace-equaling relation θ and C-kernel-equaling relation κ, respectively, are complete sublattices of C_P(S) and the greatest elements of these sublattices are given.     

正则纯整群带的算子半群和同余网

正则半群Ｓ的同余格（Ｓ）上的算子Ｋ，ｋ，Ｔ和ｔ定义如下：对于ρ∈Ｓ，ρＫ和ρｋ（ρＴ和ρｔ）分别是与ρ有相同核（迹）的最大和最小同余．我们确定了所有正则纯整群带的同余格上由Ｋ，ｋ，Ｔ和ｔ生成的算子半群．并确定了正则纯整群带上任意同余的同余网．     

关于拟P-内射模的一些注记(英文)

设 R是一个环 .一个右 R-模 M叫做拟 P-内射的 ,如果 M的每个 M-循环子模到 M的任一个 R-同态都能扩展到 M.假设 M是一个自生成子的拟 P-内射模 .在这篇文章中 ,我们表明如果这样一个模是一个 CF-模 (特别地 ,CS-模 ) ,那么 S/J(S)是正则的 ,其中 S=End(MR) .进一步 ,如果 S是半素环 ,那么 M的每个极大核是 M的一个直和项 .这些结果扩展了 P-内射环的一些结果     

P-反演半群上的正则P-同余

令S(P)为P-反演半群，本文借助于P-核正规系来刻画S(P)上的强P-同余，证明了S(P)上的任一正则P-同余可以决定S(P)的一个P-核正规系；反之，S(P)的任一P-核正规系可以决定S(P)上的一个正则P-同余．     

Spectral problem for polynomial matrix pencils. 2

For an arbitrary polynomial pencil of matrices A_i of dimensions m×n one presents an algorithm for the computation of the eigenvalues of the regular kernel of the pencil. The algorithm allows to construct a regular pencil having the same eigenvalues as the regular kernel of the initial pencil or (in the case of a dead end termination) allows to pass from the initial pencil to a pencil of smaller dimensions whose regular kernel has the same eigenvalues as the initial pencil. The problem is solved by reducing the obtained pencil to a linear one. For solving the problem in the case of a linear pencil one considers algorithms for pencils of full column rank as well as for completely singular pencils.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 111, pp. 109–116, 1981.     

P-stable models of strong kernel programs

The research community has long recognized the study of non-monotonic reasoning (NMR) as a promising approach to model features of commonsense reasoning. We study one of the semantics that are useful to formalize NMR, called the p-stable semantics. We introduce three different formats for normal programs: negative normal programs, restricted negative normal programs and strong kernel programs. These forms help to simplify the search of p-stable models of the original program. One of the main results of this paper indicates that the p-stable semantics for strong kernel programs is the same as the stable semantics. This way, all the applications based on stable semantics for those kernel programs (defined in [S. Costantini, A. Provetti, Normal forms for answer set programming, J. Theory Pract. Log. Program. 5 (2005) 747-760]) that are strong kernel programs can also be based on the p-stable semantics.     

Traveling wave solutions for Schrödinger equation with distributed delay

The paper is devoted to study of traveling waves of nonlinear Schrödinger equation with distributed delay by applying geometric singular perturbation theory, differential manifold theory and the regular perturbation analysis for a Hamiltonian system. Under the assumptions that the distributed delay kernel is strong general delay kernel and the average delay is small, we first investigate the existence of solitary wave solutions by differential manifold theory. Then by utilizing the regular perturbation analysis for a Hamiltonian system, we explore the periodic traveling wave solutions.     

Quantum double suspension and spectral triples

In this paper we are concerned with the construction of a general principle that will allow us to produce regular spectral triples with finite and simple dimension spectrum. We introduce the notion of weak heat kernel asymptotic expansion (WHKAE) property of a spectral triple and show that the weak heat kernel asymptotic expansion allows one to conclude that the spectral triple is regular with finite simple dimension spectrum. The usual heat kernel expansion implies this property. The notion of quantum double suspension of a C^?-algebra was introduced by Hong and Szymanski. Here we introduce the quantum double suspension of a spectral triple and show that the WHKAE is stable under quantum double suspension. Therefore quantum double suspending compact Riemannian spin manifolds iteratively we get many examples of regular spectral triples with finite simple dimension spectrum. This covers all the odd-dimensional quantum spheres. Our methods also apply to the case of noncommutative torus.     

Obtaining upper bounds of heat kernels from lower bounds

We show that a near‐diagonal lower bound of the heat kernel of a Dirichlet form on a metric measure space with a regular measure implies an on‐diagonal upper bound. If in addition the Dirichlet form is local and regular, then we obtain a full off‐diagonal upper bound of the heat kernel provided the Dirichlet heat kernel on any ball satisfies a near‐diagonal lower estimate. This reveals a new phenomenon in the relationship between the lower and upper bounds of the heat kernel. © 2007 Wiley Periodicals, Inc.     

Congruences on eventually regular semigroups

Let S be an eventually regular semigroup. The extensively P-partial congruence pairs and P-partial congruence pairs for S are defined. Furthermore, the relationships between the lattice of congruences on S, the lattice of P-partial kernel normal systems for S, the lattice of extensively P-partial kernel normal systems for S and the poset of P-partial congruence pairs for S are explored.     

P-systems in regular semigroups

In this paper, firstly it is shown that a regular semigroup S becomes a regular *-semigroup (in the sense of [1]) if and only if S has a certain subset called a p-system. Secondly, all the normal *-bands are completely described in terms of rectangular *-bands (square bands) and transitive systems of homomorphisms of rectangular *-bands. Further, it is shown that an orthodox semigroup S becomes a regular *-semigroup if there is a p-system F of the band E_S of idempotents of S such that F∋e, E_S∋t, e≥t imply t∈F. By using this result, it is also shown that F is a p-system of a generalized inverse semigroup S if and only if F is a p-system of F_S. Dedicated to Professor L. M. Gluskin on his 60th birthday     

平衡范畴与半群的双序

研究范畴与半群通过幂等元双序建立的一种自然联系.对每个有幂等元的半群S,其幂等元生成的左、右主理想之集通过双序ω~e,ω~r自然确定两个有子对象、有像且每个包含都右可裂的范畴L(S),R(S),其中态射的性质与S中元素的富足性、正则性有自然对应.利用这个联系,我们定义了"平衡(富足、正规)范畴"概念.对任一平衡(富足、正规)范畴■,我们构造其"锥半群"■,证明■左富足(富足、正则),且每个平衡(富足、正规)范畴■都与某左富足(富足、正则)半群S的左主理想范畴L(S)(作为有子对象的范畴)同构.     

New concepts for completely regular semigroups

Let 𝒞? denotes the variety of completely regular semigroups considered with the unary operation of inversion. The global study of the lattice of subvarieties of 𝒞? depends heavily on various decompositions. Some of the most fruitful among these are induced by the kernel and the trace relations. In their turn, these relations are induced by the kernel and the trace relations on the lattice of congruences on regular semigroups. These latter admit the concepts of kernel and trace of a congruence. The kernel and the trace relations for congruences were transferred to kernel and trace relations on varieties but the kernel and trace got no analogue for varieties.

We supply here the kernel and the trace of a variety which induce the relations of their namesake. For the local and core relations, we also define the local and core of a variety. All the new concepts are certain subclasses of 𝒞?. In this way, we achieve considerable similarity of the new concepts with those for congruences. We also correct errors in two published papers.     

A Cauchy kernel for slice regular functions

In this article, we show how to construct a regular, non-commutative Cauchy kernel for slice regular quaternionic functions. We prove an (algebraic) representation formula for such functions, which leads to a new Cauchy formula. We find the expression of the derivatives of a regular function in terms of the powers of the Cauchy kernel, and we present several other consequent results.