An inversion formula for the primitive idempotents of the trivial source algebra

Formulas for the primitive idempotents of the trivial source algebra, in characteristic zero, have been given by Boltje and Bouc–Thévenaz. We shall give another formula for those idempotents, expressing them as linear combinations of the elements of a canonical basis for the integral ring. The formula is an inversion formula analogous to the Gluck–Yoshida formula for the primitive idempotents of the Burnside algebra. It involves all the irreducible characters of all the normalizers of p-subgroups. As a corollary, we shall show that the linearization map from the monomial Burnside ring has a matrix whose entries can be expressed in terms of the above Brauer characters and some reduced Euler characteristics of posets.     

Triangular Matrix Representations

In this paper we develop the theory of generalized triangular matrix representation in an abstract setting. This is accomplished by introducing the concept of a set of left triangulating idempotents. These idempotents determine a generalized triangular matrix representation for an algebra. The existence of a set of left triangulating idempotents does not depend on any specific conditions on the algebras; however, if the algebra satisfies a mild finiteness condition, then such a set can be refined to a “complete” set of left triangulating idempotents in which each “diagonal” subalgebra has no nontrivial generalized triangular matrix representation. We then apply our theory to obtain new results on generalized triangular matrix representations, including extensions of several well known results.     

Idempotents in the Endomorphism Ring of an Ideal in a p-Extension of a Complete, Discrete Valuation Field with Residue Field of Characteristic p as a Galois Module

In this paper, the question concerning the existence of nontrivial idempotents in the endomorphism ring of an ideal in a p-extension of a complete discrete valuation field with residue field of characteristic p > 2 as a Galois module is considered. The nonexistence of nontrivial central idempotents for a non-Abelian totally widely ramified extension is proved. Bibliography: 4 titles.     

On the question of embedding a semigroup into a semiband

In this paper we present a new embedding of a semigroups into a semiband (idempotent-generated semigroup) of depth 4 (every element is the product of 4 idempotents) using a semidirect product construction. Our embedding does not assume that S is a monoid (although it assumes a weaker condition) and works also for (non-monoid) regular semigroups. In fact, this semidirect product construction works particularly well if the semigroup is regular: we can choose the semiband to have depth 2. We shall see that many properties of S are preserved by this construction and we shall compare it to other known embeddings.     

A Compact Monothetic Semitopological Semigroup Whose Set of Idempotents Is Not Closed

For each monothetic unitary group without proper co-compact subgroups, we show the existence of a compact semitopological semigroup (affine) compactification in which the set of idempotents is not closed answering a question of Berglund.     

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     

Group invariant weighing matrices

We investigate the existence problem of group invariant matrices using algebraic approaches. We extend the usual concept of multipliers to group rings with cyclotomic integers as coefficients. This concept is combined with the field descent method and rational idempotents to develop new non-existence results.     

Note on exclusive semigroups

In the SEMIGROUP FORUM, Vol. 1, No. 1, B. M. Schein proposed the following problem: Describe the structure of semigroups S such that for every a,b,c∈S, abc=ab, bc or ac. At present, we shall call such a semigroup S anexclusive semigroup. Recently, the author heard that the structure of commutative exclusive semigroups was completely determined by T. Tamura [3]. In this paper, we deal with exclusive semigroups which are not necessarily commutative. The paper is divided into three sections. At first, the structure of exclusive semigroups whose idempotents form a rectangular band will be clarified. Next, we shall investigate a certain class of exclusive semigroups called “exclusive homobands”. Especially, in the final section we shall deal with medial exclusive homobands and show how to construct them. The proofs are omitted and will be given in detail elsewhere.     

On the identities in algebras generated by linearly connected idempotents

We investigate the problem of the existence of polynomial identities (PI) in algebras generated by idempotents whose linear combination is equal to identity. In the case where the number of idempotents is greater than or equal to five, we prove that these algebras are not PI-algebras. In the case of four idempotents, in order that an algebra be a PI-algebra, it is necessary and sufficient that the sum of the coefficients of the linear combination be equal to two. In this case, these algebras are F₄-algebras.Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 6, pp. 782–795, June, 2004.     

一类替换环的偏序$K_{0}$-群的无扭性

A ring R is called orthogonal if for any two idempotents e and f in R, the condition that e and f are orthogonal in R implies the condition that [eR] and [fR] are orthogonal in K0（R）^＋, i.e., [eR]∧[fR] = 0. In this paper, we shall prove that the K0-group of every orthogonal, IBN2 exchange ring is always torsion-free, which generalizes the main result in [3].     

Generators for categories of compact irreducible semigroups

For a compact totally ordered space X, K. H. Hofmann and P. S. Mostert constructed a topological semigroup Irr (X)₀ such that every compact irreducible semigroup with idempotents X is a surmorphic image of Irr (X)₀ [2]. J. H. Carruth and M. Mislove pointed out that Irr (X)₀ was in general not a compact semigroup. In this paper a compact connected hormos will be constructed which contains Irr (X)₀ as a dense subsemigroup and for which every compact irreducible semigroup with idempotents X is a surmorphic image. This leads to a new proof of the existence of generators for the category of compact irreducible semigroups with idempotents X. It will then be shown that Irr (X)₀ contains generators for the category.     

On semigroups admitting ring structure II

We shall consider semigroups with O, which contain a unique maximal right ideal generated by a finite number of independent generators and in which every proper right ideal is contained in the unique maximal right ideal and investigate when these semigroups are multiplicative semigroups of a ring. We prove in particular that the necessary condition for this class of semigroups S to admit ring structure is S=S² if |S|>2. Furthermore the admissible ring structure of S is determined when the product of every two generators of the maximal right ideal M is O and when S satisfies one of the two conditions, namely S is commutative without idempotents except O and 1 or every generator of M is nilpotent.     

Characterizations and representations of the group inverse involving idempotents

This paper is to present some results on the group invertibility of products and differences of idempotents. In addition, some formulae for the group inverse of sums, differences and products of idempotents are established by using some given idempotents.     

关于幂等元之差的可逆性

本文研究在一个有单位元的环中两个幂等元之差的可逆性问题,利用幂等元的性质,得到了两个幂等元之差可逆的几个充分必要条件,并给出了在矩阵环中的几个应用.     

具有弱中间幂等元的正则半群

本文在正则半群上引入弱中间幂等元和拟中间幂等元,着重探讨了这两类幂等元的性质特征.构造了若干具有弱(拟)中间幂等元的正则半群,确定了弱中间幂等元和拟中间幂等元之间的关系,给出了弱中间幂等元和拟中间幂等元各自的等价判定,利用拟中间幂等元刻画了纯正半群.最后,得到了构造具有拟中间幂等元的正则半群的一般途径,并在此基础上进一步给出了判定正则半群是否具有乘逆断面的方法.     

具有弱正规幂等元的富足半群的结构

本文研究含弱正规幂等元的富足半群.在给出这类半群的若干特征后,建立了具有弱正规幂等元的富足半群的结构．作为应用,给出具有正规幂等元的富足半群和具有（弱）正规幂等元的拟适当半群的结构．     

A Decomposition of the Descent Algebra of a Finite Coxeter Group

The purpose of this paper is twofold. First we aim to unify previous work by the first two authors, A. Garsia, and C. Reutenauer (see [2], [3], [4], [5] and [10]) on the structure of the descent algebras of the Coxeter groups of type A _n and B _n. But we shall also extend these results to the descent algebra of an arbitrary finite Coxeter group W. The descent algebra, introduced by Solomon in [14], is a subalgebra of the group algebra of W. It is closely related to the subring of the Burnside ring B(W) spanned by the permutation representations W/W _J, where the W _J are the parabolic subgroups of W. Specifically, our purpose is to lift a basis of primitive idempotents of the parabolic Burnside algebra to a basis of idempotents of the descent algebra.     

Invariance under outer inverses

We shall use the minus partial order combined with Pierce’s decomposition to derive the class of outer inverses for idempotents, units and group invertible elements. Subsequently we show, for matrices over a field \({\mathbb {F}}\), that the triplet \(B\hat{A}C\) is invariant under all choices of outer inverses of A if and only if \(B = 0\) or \(C = 0\).     

Équation Périodique de Navier–Stokes sans Viscosité dans une Direction

ABSTRACT

We study the periodic Navier–Stokes equation when the vertical viscosity vanishes, in a critical space (invariant by scaling). We shall prove local-in-time existence of the solution. When the tridimensional part of the initial data is small compared with the bidimesional part and with the horizontal viscosity, we shall show global existence of solutions.