Showing papers on "Idempotence published in 1993"

Products of idempotent endomorphisms of an independence algebra of infinite rank

Abstract: BY JOHN FOUNTAI ANND ANDREW LEWINDepartment of Mathematics, University of York, Heslington, York YOl 5DD(Received 12 August 1992; revised December 22 1992)AbstractIn 1966, J. M. Howie characterized the self-maps of a set which can be written as aproduct (under composition) of idempotent self-maps of the same set. In 1967, J. A.Erdos considered the analogous question for linear maps of a finite dimensionalvector space and in 1985, Reynolds and Sullivan solved the problem for linear mapsof an infinite dimensional vector space. Using the concept of independence algebra,the authors gave a common generalization of the results of Howie and Erdos for thecases of finite sets and finite dimensional vector spaces. In the present paper weintroduce strong independence algebras and provide a common generalization of theresults of Howie and Reynolds and Sullivan for the cases of infinite sets and infinitedimensional vector spaces.IntroductionFor a mathematical structure M the set of endomorphisms oiM, which we denoteby End(.M), is a monoid under composition of mappings. We let E denote the set ofnon-identity idempotents of End(M). Over the last twenty five years considerableeffort has been devoted to describing the subsemigroup (Ey generated by E. The firstresults were obtained by Howie in [5] where a set-theoretic description of (E} isgiven when M is simply a set and End(ilf) = T(M) is the full transformationsemigroup on M. To describe when if is an infinite set, we define, for a in T(M),C(oc) = {xeM:\(xcc)a-

Combination techniques and decision problems for disunification

Abstract: Former work on combination techniques was concerned with combining unification algorithms for disjoint equational theories E 1,..., E n in order to obtain a unification algorithm for the union E 1, ∪...∪ E n of the theories. Here we show that variants of this method may be applied to disunification as well. Solvability of disunification problems in the free algebra of the combined theory E 1 ∪...∪ E n is shown to be decidable if solvability of disunification problems with linear constant restrictions in the free algebras of the theories E i is decidable. In order to decide ground solvability (i.e., solvability in the initial algebra) of disunification problems in E 1 ∪...∪ E n we have to consider a new kind of subproblem for the particular theories E i , namely solvability (in the free algebra) of disunification problems with linear constant restriction under the additional constraint that values of variables are not E i -equivalent to variables. The correspondence between ground solvability and this new kind of solvability holds, (1) if one theory E i is the free theory with at least one function symbol and one constant, or (2) if the initial algebras of all theories E i are infinite. Our results can be used to show that the existential fragment of the theory of the (ground) term algebra modulo associativity of a finite number of function symbols is decidable; the same result follows for function symbols which are associative and commutative, or associative, commutative and idempotent.

Proper left type- A monoids revisited

Abstract: The relation ℛ* is defined on a semigroup S by the rule that ℛ* b if and only if the elements a, b of S are related by the Green's relation ℛ in some oversemigroup of S . A semigroup S is an E -semigroup if its set E(S) of idempotents is a subsemilattice of S . A left adequate semigroup is an E -semigroup in which every ℛ*-class contains an idempotent. It is easy to see that, in fact, each ℛ*-class of a left adequate semigroup contains a unique idempotent [2]. We denote the idempotent in the ℛ*-class of a by a + .

On idempotence and related requirements in edge detection

Abstract: The energy feature detectors described by R. Owens et al. (1989) are good candidates for idempotent edge detectors. However, some of them (in particular, the Gabor energy feature detector) suffer from a serious defect that is absent in gradient-type operators: their sensitivity to gray-level shift in the original image. This leads to errors in the localization of step edges. The Fourier phase and amplitude conditions outlined by M.C. Morrone and D.C. Burr (1988) for the class of energy feature detectors guarantee a zero DC level when the convolution masks are taken in L/sup 1/; therefore, the resulting energy feature detector is invariant under grey-level shift in the original image. Also, the properties of the underlying edge model are invariant under a smoothing of the image by a Gaussian or any function in L/sup 1/ having zero Fourier phase. In particular, such a smoothing does not deteriorate the idempotence of the edge detector. Some concrete examples of energy feature detectors satisfying the Morrone conditions are described. >

The complete reduction of Clifford algebras

Abstract: The complete reduction of simple Clifford algebras is spanned by matrix bases. Considered as generators of an algebra the basis elements pij satisfy the product relations pijpkl= δjkpil. It follows that subsets of pij with subscript or superscript fixed span minimal left or right ideals. The expression of the basis elements pij in the usual multivector basis is derived from isomorphic matrix algebras. Spinors and Hermitian conjugate spinors are defined as members of minimal left and right ideals. Spinor scalar products are defined, these scalar products are in fact scalar multiples of an idempotent.

Idempotent and compact matrices on linear lattices: a survey of some lattice results and related solutions of finite relational equations

Abstract: After a survey of some known lattice results, we determine the greatest idempotent (resp. compact) solution, when it exists, of a finite square rational equation assigned over a linear lattice. Similar considerations are presented for composite relational equations.

Quasi-heredity of algebras and their factor algebras

Abstract: Let A be a finite-dimensional algebra over an algebraically closed field and denote by N the Jacobson radical of A. If there is an integer i > 2 such that A/Ni is quasi-hereditary, then A is quasi-hereditary. Let A be a finite-dimensional algebra over an algebraically closed field k. By N we denote the Jacobson radical of A. An ideal of A is called a heredity ideal of A if it satisfies (1) J2 = J, (2) JNJ = 0, and (3) J is a projective left A-module. We recall that the algebra A is said to be quasi-hereditary provided there is a chain O c J1 c J2 c c Jn = A of ideals of A such that JilJi-l is a heredity ideal of A/Ji-I for all i = 1, . .. , n . Some basic properties on quasi-hereditary algebras may be found in [DR]. The aim of this note is to show the following: If the algebra A is not quasi-hereditary, then, for any i > 2, the factor algebra A/Ni never becomes a quasi-hereditary algebra. Throughout this note all algebras are finite-dimensional k-algebras with 1, module means finitely generated left module. By a (or J) we denote the image of a E A (or J c A) under the canonical map A -* A/I, where I is an ideal of A. The above-mentioned result may be reformulated as the following theorem. Theorem 1. Let A be a basic connected algebra with Jacobson radical N. If A/Ni is quasi-hereditary for some i > 2, then A is quasi-hereditary. To prove this result we need some preparations. Lemma 2. Let A be a basic algebra and e be a primitive idempotent such that J = AeA is a heredity ideal of A. Then eAe k. Proof. Since the field k is algebraically closed and the Jacobson radical of eAe is eNe, it follows from the definition of a heredity ideal that eNe = 0 and eAek. Received by the editors September 18, 1990 and, in revised form, March 27, 1992. 1991 Mathematics Subject Classification. Primary 16P20, 16G10.