Showing papers on "Idempotence published in 1992"

Homological theory of idempotent ideals

Abstract: Let Λ be an artin algebra U and a two-sided ideal of Λ. Then U is the trace of a projective Λ-module P in Λ. We study how the homological properties of the categories of finitely generated modules over the three rings Λ/U, Λ and the endomorphism ring of P are related. We give some applications of the ideas developed in the paper to the study of quasi-hereditary algebras

A new set of fast algorithms for mathematical morphology I: idempotent geodesic transforms

Abstract: This paper generalizes the geodesic transforms used in mathematical morphology for binary image processing by introducing the concept of morphological operations constrained by an initial set and a transformable set. These new operations allow the efficient extension of geodesic constraints to grayscale image processing as they can be implemented with simple algorithms. They also make possible the application of new constraints, leading to new transforms with new properties. Two particular cases of these new transforms are then studied thoroughly: erosion and dilation operations constrained by an initial set and a transformable set that are iterated until idempotence is reached, or idempotent geodesic transforms (IGT). A sequential local transform (SLT) algorithm is presented that greatly speeds up the computation of IGT; the conditions for convergence for the algorithm and the relation between speed of convergence and the complexity of the initial set and the transformable set are investigated.

Endomorphisms of semimodules over semirings with an idempotent operation

Abstract: For an arbitrary endomorphism of the free semimodule over an Abelian semiring with operations and it is shown under the assumption that is idempotent (and under certain other restrictions on ) that there exists a nontrivial "spectrum", i.e., there exist a and a nontrivial subsemimodule such that for any . The same result is also obtained for endomorphism analogues of integral operators (in the sense of the theory of idempotent integration). In terms of this spectrum investigations are made of the asymptotic behavior of endomorphisms under iteration and of convergence of the "Neumann series" appearing in the solution of the equations . The simplest examples are connected with the semiring and arise, for example, in dynamic programming problems.

On bernstein algebras which are train algebras

Abstract: holds in A. This class of algebras was introduced by Holgate [4], following the original work of Bernstein [2] and subsequent investigations by Lyubich [5] on idempotent quadratic maps from a real simplex into itself. A summary of known results on Bernstein algebras (up to 1980) is given in Worz-Busekros [8], which will also be used as a basic reference on algebras in genetics. All definitions not explicitly stated here can be found in this monograph. Bernstein algebras are not necessarily genetic algebras in the sense of Schafer [6]; indeed there are Bernstein algebras which are not even train algebras, cf. Worz-Busekros [8]. In this note Bernstein algebras which are train algebras are considered. K is assumed to be of characteristic zero throughout, although this hypothesis could be weakened for some of the results. As follows from a result of Outtara [9], the rank polynomial of a Bernstein train algebra is uniquely determined by its degree. This result admits an interpretation as a "stationarity principle" which is different from (1), and this in turn enables us to solve the differential equation for overlapping generations in the time-continuous model (for K = IR) in closed form. In particular, the long-term behaviour of the solutions can easily be determined. I thank the referee for valuable comments and suggestions.

Calculating the Warshall/Floyd path algorithm

Abstract: A calculational derivation is given of an all· pairs path algorithm two instances of which are Warshall's reach ability algorithm and Floyd's shortest-path algorithm. The derivation provides an elementary example of the importance of the so-called star-decomposition rule. 1 Algebraic Framework This paper presents a calculational derivation of an all-pairs path algorithm, two well-known instances of which a.re Warshall's (reachability) algorithm and Floyd's shortest-path algorithm. The calculations presented here are essentially the same as those in [1, 2]. The presentation has been brought up-to-date in that explicit rather than implicit use is made of invariant properties. Moreover notational refinements enhance the clarity of the derivation. Like [1, 2] the framework for the current derivation is regular algebra. The axioms of regular a.lgebra the a.lgebra of regula.r languages are now widely known a.nd publicised. (See e.g. [6, 4].) The fact that the elementa.ry operators involved in several path-finding algorithms obey the a.xioms of regular algebra. is also widely known and this knowledge will be assumed. The specific details of the framework are that (S, +,·,*,0,1) is a regular algebra. That is, S is a set on which are defined two binary operators + and· and one unary operator' (written as a postfix of its argument). Addition (+) is associative, commutative and idempotent. Multiplication (-) distributes over addition and is associat.ive but is not necessarily commutative. The basic properties of ' that we use here are, for all a, b E S : (1) a . =1+(1'(1 * (2) a . (b . a)* = (a W . a (3) (a + b r = (a' . br * . a

The structure of the set of idempotents in a Banach algebra

Abstract: We study here the algebraic, geometric, and analytic structure of the set of idempotent elements in a real or complex Banach algebra. A neighborhood of each idempotent in the set of idempotents forms the set of idempotents in a Rees product subsemigroup of the Banach algebra. Each nontrivial connected component of the set of idempotents is shown to be a generalized saddle, a type of analytic manifold. Each component is also shown to be the quotient of a (possibly infinite dimensional) Lie group by a Lie subgroup.

Logical context of nonlinear filtering

Abstract: The mathematical structure of binary nonlinear filtering is expressed in the context of binary cellular logic and the relevance of existing image algebras is discussed. Operator properties such as antiextensively and idempotence are examined from a discrete logical perspective, as are the classical Matheron representations. The simplicity of the operational properties is exposed by such an approach, as is the use of commonplace logic design methods for the composition and decomposition of nonlinear filters, in particular, binary morphological filters.

Lattice of idempotent separating congruences in a ℙ-regular semigroup

Abstract: In this paper we introduce the concept of ℙ-regular, normal subset of a ℙ-regular semigroup, give an alternate characterisation of the maximum idempotent separating congruence in a ℙ-regular semigroup and finally describe the lattice of the idempotent separating congruences in a ℙ-regular semigroup.

On some problems of Petrich concerning Bruck and Reilly semigroups

Abstract: For an inverse semigroupS with its idempotents dually well-ordered, we prove thatS is isomorphic to the semigroup of all one-to-one partial right translations ofS. Also, we prove for a Bruck semigroupS=B(T, α) thatS isE-unitary if and only ifT isE-unitary and α is an idempotent pure homomorphism. Moreover, we characterize allE-unitary covers ofB(T, α), whereT is a finite chain of groups.

Idempotent comultiplications on graded algebras

Abstract: We classify all idempotent comultiplications on a graded anticommutative algebra up to degree 3, provided its components are torsion free, and topologically realize all algebraic possibilities. Then we extend some results to dimension n and obtain topological consequences about closed n-manifolds with cohomology of special type.