Showing papers on "Idempotence published in 2000"

On structures preserved by idempotent transformations of groups and homotopy types

Abstract: Recent work on homotopical localizations has revealed that many structures and properties are preserved under idempotent functors in the cat- egory of groups and in the homotopy category of spaces. In this article we collect a number of findings in this direction, including new results by the au- thor and coauthors. Background and proofs are given throughout. We discuss in particular the preservation of asphericity of spaces under localization at sets of primes, aiming to the study of localizations of infra-nilmanifolds.

Generalized Idempotence in Fuzzy Mathematical Morphology

Abstract: In binary mathematical morphology, the idempotence of the binary closing and binary opening has led to the study of objects that are closed or open w.r.t. a given structuring element. For these objects, an interesting generalized idempotence law holds. In recent years, fuzzy mathematical morphology has been proposed as an alternative theory to gray-scale morphology. In this paper, we consider the most general approach, the so-called logical approach, based on fuzzy logical operators. In particular, the role of t-norms and their residual operators is stressed. It is shown that the generalized idempotence laws still hold in fuzzy mathematical morphology when choosing a continuous t-norm and its residual implicator as underlying conjunctor and implicator, and in particular when choosing a nilpotent t-norm and its residual implicator. In the latter case, the fuzzy closing and fuzzy opening are dual operations, and hence results concerning the fuzzy closing are obtained by duality from the fuzzy opening. The unique role of nilpotent t-norms, and hence of the Lukasiewicz t-norm, is demonstrated.

Groups in the class semigroup of a prüfer domain of finite character

Abstract: The class semigroup of a commutative integral domain R is the semi­group S(R) of the isomorphism classes of the nonzero ideals of R with operation induced by multiplication. We consider Prufer domains of finite character, i.e. Prufer domains in which every nonzero ideal is contained but in a finite number of maximal ideals. In [1] it is proved that, if R is such a Prufer domain, then the semigroup S(Ris a Clifford semigroup, namely it is the disjoint union of the subgroups associated to each idempotent element. In [2] we gave a description of a generating set for the A-semilattice of the idempotent elements of S(R). In this paper we consider the constituent groups of the class semigroup. We prove that the groups associated to idempotent elements of S(R) are extensions of class groups of overrings of (R) by means of direct products of archimedean groups of localizations of(R) at idempotent prime ideals.

A Class of Idempotent Semirings

Abstract: -relations of the two reducts. We identify important subvarieties and give structure theorems for special types of such semirings.

The grothendieck group of the category of modules of finite projective dimension over certain weakly triangular algebras

Abstract: In this paper we study the category of finitely generated modules of finite projective dimension over a class of weakly triangular algebras, which includes the algebras whose idempotent ideals have finite projective dimension. In particular, we prove that the relations given by the (relative) almost split sequences generate the group of all relations for the Grothendieck group of P <∞(Λ) if and only if P <∞(Λ) is of finite type. A similar statement is known to hold for the category of all finitely generated modules over an artin algebra, and was proven by C.M.Butler and M. Auslander ( [B] and [A]).

The radical in alternative baric algebras

Abstract: In this paper we proved that in a baric, power associative, F-algebra of finite dimension, there is an idempotent element of weight 1. When this algebra is alternative, with this idempotent we make the Peirce's decomposition. We define the nil radical as the maximal nil ideal. We proved that the bar-radical is the intersection of the nil radical with the square of \(bar({\cal a})\). All the results are naturally true for baric associative algebras of finite dimension.

Moore--Penrose Inverse of Matrices on Idempotent Semirings

Abstract: We characterize matrices over an idempotent semiring satisfying some additional necessary conditions for which the Moore--Penrose inverse exists. The "(max,×) semiring," defined as the set of nonnegative real numbers ${\mathbb R}^+,$ equipped with the operations $a\oplus b= \max\{a,b\}\ \mbox{and}\ a\otimes b= ab$ is an example of such a semiring. The "(max,×) semiring,"', defined as the set of real numbers including $-\ity,$ equipped with the operations $a\oplus b= \max\{a,b\}\ \mbox{and}\ a\otimes b= a+b$ is another example. Some of our results generalize known results in the case of the binary boolean algebra (a trivial idempotent semiring). We give an algorithm to compute the Moore--Penrose inverse, when it exists. We also make comparisons with similar results over the conventional algebra.

Idempotent functional analysis: an algebraic approach

Abstract: In this paper we consider Idempotent Functional Analysis, an `abstract' version of Idempotent Analysis developed by V. P. Maslov and his collaborators. We give a review of the basic ideas of Idempotent Analysis. The correspondence between concepts and theorems of the traditional Functional Analysis and its idempotent version is discussed; this correspondence is similar to N. Bohr's correspondence principle in quantum theory. We present an algebraical approach to Idempotent Functional Analysis. Basic notions and results are formulated in algebraical terms; the essential point is that the operation of idempotent addition can be defined for arbitrary infinite sets of summands. We study idempotent analogs of the main theorems of linear functional analysis and results concerning the general form of a linear functional and scalar products in idempotent spaces.

Trace methods in twisted group algebras, II

Abstract: In this note, we continue our discussion of trace methods in twisted group algebras. Specifically, we obtain the twisted analog of Bass' theorem on the traces of idempotents in ordinary group algebras. Indeed, we show that with suitable normalization, the characteristic 0 trace values of an idempotent are all contained in a cyclotomic field. The proof is a variant of the original argument combined with a reduction to finitely presented groups.

Matching modulo associativity and idempotency is NP-complete

Abstract: We show that AI-matching (AI denotes the theory of an associative and idempotent function symbol), which is solving matching word equations in free idempotent semigroups, is NP-complete. Note: full version of the paper appears as [8].

A characterisation of a class of semigroups with locally commuting idempotents

Abstract: McAlister proved that a necessary and sufficient condition for a regular semigroup S to be locally inverse is that it can be embedded as a quasi-ideal in a semigroup T which satisfies the following two conditions: (1) T = TeT, for some idempotent e; and (2) eTe is inverse We generalise this result to the class of semigroups with local units in which all local submonoids have commuting idempotents

Sheaves on quantaloids

Abstract: We generalize the well-known notions of a singleton, completeQ-set, presheaf, and sheaf over a complete Heyting algebra or a right-sided idempotent quantale to arbitrary quantaloids. We show that every completeQ-set can be viewed as a sheaf and vice versa.

Achieving idempotence in near-lossless JPEG-LS

Abstract: The lossless and near-lossless image compression standard, JPEG-LS, while offering state-of-the-art compression performance with low complexity, fails to be idempotent in near-lossless mode (i.e., images degrade upon successive compression/decompressions). This paper identifies the cause and presents two solutions. First it presents a modification to the compressor and decompressor that maintains or improves the error bounds and achieves idempotence. Second it describes a preprocessor that acts upon any image and returns one upon which JPEG-LS does perform idempotently at the expense of doubling the guaranteed error bound on a small subset of pixels (typically below 0.5%).

Topology of the Semigroup of Singular Endomorphisms

Abstract: of singular endomorphisms of an n-dimensional vector space over K are discussed here. Since S n is known to be an idempotent generated regular semigroup, we pay more attention to the topological properties of the set E n of idempotents in S n . The local structure of E n is shown to be that of a C infinity-manifold and of a finite-dimensional vector bundle over the Grassmann manifolds. The topology of the biorder relations and sandwich sets are also discussed.

On Embeddability of Idempotent Separating Extensions of Inverse Semigroups

Abstract: be an inverse semigroup and rho an idempotent separating congruence on S. It is proved that S can be embedded into a lambda-semidirect product of a group F by S/rho where F belongs to the variety generated by the idempotent classes of rho.

Matching Modulo Associativity and Idempotency is NP-Complete

Abstract: We show that AI-matching (AI denotes the theory of an associative and idempotent function symbol), which is solving matching word equations in free idempotent semigroups, is NP-complete. Note: this is a full version of the paper [9] and a revision of [8].

Banach algebras generated by two idempotents and one flip

Abstract: The paper is concerned with the invertibility of elements in a Banach algebra generated by two idempotents and one flip element. A symbol in form of a 2 × 2 matrix function which is defined on some Hausdorff compact will be constructed and a complete description of will be given.

The Thick Subcategory Generated by the Trivial Module

Abstract: This paper is a report of my lecture at the International Conference on the Representation Theory of Algebras which preceded the Euroconference on Infinite Length Modules. While it is true that the aim of the research is the homological algebra of finitely presented modules, a major point of the lecture was the role that infinitely generated modules could play in the investigation. The methods of idempotent modules and the theory of support varieties for infinite dimensional modules have already had a significant impact on group representation theory. It seems certain that there will be a lot more to follow. I am honored by the invitation to include the report in the conference proceedings, and I would like to thank the organizers of the conference and the workshop for the stimulating experience.

The arithmetic of a semigroup of series of Walsh functions

Abstract: Let be the classical system of the Walsh functions, the multiplicative semigroup of the functions represented by series of functions Wk(t)with non-negative coefficients which sum equals 1. We study the arithmetic of . The analogues of the well-known [ related to the arithmetic of the convolution semigroup of probability measures on the real line are valid in . The classes of idempotent elements, of infinitely divisible elements, of elements without indecomposable factors, and of elements without indecomposable and non-degenerate idempotent factors are completely described. We study also the class of indecomposable elements. Our method is based on the following fact: is isomorphic to the semigroup of probability measures on the groups of characters of the Cantor-Walsh group.

Biorder Structure of the Bilinear Form Semigroup

Abstract: If X and Y are two-finite dimensional vector spaces over a field K and B: X × Y → K is a bilinear form, in [6], we have established that the set of adjoint pairs of linear maps forms a regular subsemigroup of L(X) × L(Y) op . We call it the bilinear form semigroup. Here, we see that the maximal {\cal D} class in this semigroup is a Completely Simple Orthodox semigroup. Again, we note that given any idempotent. We can find a maximal covering chain containing it and ending at a maximal idempotent. We use this to obtain an estimate for the degeneracy of the bilinear form in terms of the rank of maximal idempotents.

Fully invariant transformations and associated groups

Abstract: It is well known that the symmetric group S ntogether with one idempotent of rank n- 1 on a finite n-element set Nserves as a set of generators for the semigroup T nof all the total transformations on N. It is also well known that the singular part Sing n of T n can be generated by a set of idempotents of rank n- 1. The purpose of this paper is to begin an investigation of the way in which Singnand its subsemigroups can be generated by the conjugates of a subset of elements of T n by a subgroup of S n . We look for the smallest subset of elements of T n that will serve and, correspondingly, for a characterization of those subgroups of S n that will serve. Using some techniques from graph theory we prove our main result:the conjugates of a single transformation of rank n- 1 under Gsuffice to generate Singnif and only if Gis what we define to be a 2-block transitive subgroup of S n .

On supermatrix idempotent operator semigroups

Abstract: One-parameter semigroups of antitriangle idempotent supermatrices and corresponding superoperator semigroups are introduced and investigated. It is shown that $t$-linear idempotent superoperators and exponential superoperators are mutually dual in some sense, and the first gives additional to exponential solution to the initial Cauchy problem. The corresponding functional equation and analog of resolvent are found for them. Differential and functional equations for idempotent (super)operators are derived for their general $t$ power-type dependence.