Idempotence

About: Idempotence is a research topic. Over the lifetime, 1860 publications have been published within this topic receiving 19976 citations. The topic is also known as: idempotent.

CiME: Completion Modulo E

Abstract: Completion is an algorithm for building convergent rewrite systems from a given equational axiomatization. The story began in 1970 with the well-known KnuthBendix completion algorithm [8]. Unfortunately, this algorithm was not able to deal with simple axioms like commutativity (z + y = y + x) because such equations cannot be oriented into a terminating rewrite system. This problem have been solved by the so-called AC-completion algorithm of Lankford and Ballantyne [9] and Peterson and Stickel [14], which is able to deal with any permutative axioms, the most popular being assoeiativity and commutativity. In 1986, Jouannand and Kirchner [6] introduced a general T-completion algorithm which was able to deal with any theory T provided that T-congruence classes are finite, and in 1989, Bachmair and Dershowitz extended it to the case of any T such that t h e subterm relation modulo T is terminating. Because of these restrictions, these algorithms are not able to deal with the most interesting cases, AC plus unit (z + 0 = x denoted ACU) being the main one. The particular case of ACU has been investigated first in 1989 by Peterson, Baird and Wilkerson [1]: they used constrained rewriting to avoid the non-termination problem; and an ACU-completion algorithm has been described then by Jouannand and March~ in 1990 [7]. Independently from this story, in the domain of computer algebra, an algorithm for computing Gr6bner bases of polynomial ideals has been found by Buchberger in 1965 [3] and much later than that, in 1981, Loos and Buchberger [11, 4] remarked that this algorithm and the previous completion algorithms behave in a very similar way. The problem of unifying these two algorithms into a common general one arised. In 1993, using the ideas introduced for ACU-completion, Marchd described a new completion algorithm based on a variant of rewriting modulo T: normalized rewriting [12, 13], where terms have to be normalized with respect to a convergent rewrite system S equivalent to T. Of course, this assumes the existence of such an S, but this appears to be true for the examples we were interested in: AC plus unit, AC plus idempotence (x + x = x), nilpotence (x + x = 0), Abelian group theory, commutative ring theory, Boolean ring theory, finite fields theory.

Solution of generalized linear vector equations in idempotent algebra

Abstract: The problem on the solutions of homogeneous and nonhomogeneous generalized linear vector equations in idempotent algebra is considered. For the study of equations, an idempotent analog of matrix determinant is introduced and its properties are investigated. In the case of irreducible matrix, existence conditions are found and the general solutions of equations are obtained. The results are extended to the case of arbitrary matrix. As a consequence the solutions of homogeneous and nonhomogeneous inequalities are presented. 1. Introduction. For analysis of different technical, economical, and engineering systems the problems are often occurred which require the solution of vector equations linear in a certain idempotent algebra [1–5]. As a basic object of idempotent algebra one usually regards a commutative semiring with an idempotent summation, a zero, and a unity. At once many practical problems give rise to idempotent semiring, in which any nonzero (in the sense of idempotent algebra) element has the inverse one by multiplication. Taking into account a group property of multiplications, such a semiring are called sometimes idempotent semifield. Note that in passing from idempotent semrings to semifields, the idempotent algebra takes up an important common property with a usual linear algebra. In this case it is naturally expected that the solution of certain problems of idempotent algebra can be obtained by a more simple way and in a more conventional form, in particular, due to the applications of idempotent analogs of notions and results of usual algebra. Consider, for example, the problem on the solution with respect to the unknown vector x the equation A ⊗x ⊕b = x, where A is a certain matrix, b is a vector, ⊕ and ⊗ are the signs of operations of summation and multiplication of algebra. Different approaches to the solution of this equation were happily developed in the work [3–7] and the others. However many of these works consider a general case of idempotent semiring and, therefore, the represented in them results have often too general theoretical nature and are not always convenient for practical application. In a number of works it is mainly considered existence conditions of solution of equations and only some its partial (for example, minimal) solution is suggested in explicit form. In the present work a new method for the solution of linear equations in the case of idempotent semiring with the inverse one by multiplication (a semifield) is suggested which can be used for obtaining the results in compact form convenient for as their realization in the form of computational procedures as a formal analysis. For the proof of certain assertions the approaches, developed in [1, 2, 4], are used. In the work there is given first a short review of certain basic notions of idempotent algebra [2, 4, 5, 8], involving the generalized linear vector spaces and the elements of matrix calculus, and a number of auxiliary

Range projections of idempotents in C*-algebras

Abstract: In this paper we study range projections of idempotents in C*-algebras, and use them to obtain a Schur type decomposition that leads to simple proofs of results on Moore-Penrose inverse and norms of idempotents. We analyze the continuity of range projections, obtain a general result on their approximation, and recover a result of Vidav on two projections in a Hilbert space. Several representations of range projections are given. 1. Range projections Basic facts about C*-algebras needed in this paper can be found, for example, in Davidson's monograph [3]. In this paper, 21 is a unital C*algebra with unit I . By 2 l 1 we denote the set of all invertible elements of 21. We recall that I + A*A € 21\" for all A € 21 and that \\\\A*A\\\\ = \\\\A\\\\ (the C*-identity). An element A € 21 is polar if 0 is at most a pole of the resolvent of A, and quasipolar if 0 is an isolated singularity of the resolvent of A. We will need the following characterization of quasipolar elements of 21. LEMMA 1.1 [9, Theorem 4.2]. An element A G 21 is quasipolar if and only if there exists an idempotent P € 21 commuting with A such that AP is quasinilpotent and A + P E 2 l 1 . Such idempotent is unique, and is called the spectral idempotent of A corresponding to 0, written A. From the preceding lemma it follows that A is polar if and only it is quasipolar and AA = 0 for some integer k. In particular, A is simply polar if and only if A is quasipolar and AA = 0. The word 'projection' will be reserved for an element Q of a C*-algebra 21 which is self-adjoint and idempotent, that is, Q* = Q = Q. A motivation for 1991 Mathematics Subject Classification: 46L05, 46H30, 47A60.