Showing papers on "Idempotence published in 1994"

On rational series in one variable over certain dioids

Abstract: We give a characterization of rational series in one variable over certain idempotent semirings (commutative dioids) such as for instance the ``$(\max,+)$ ''semiring. We show that a series is rational iff it is merge of ultimately geometric series. As a by-product, we obtain a new proof of the periodicity theorem for powers of irreducible matrices and also some more general auxiliary results. We apply this characterization of rational series to the minimal realization problem for which we obtain an upper bound. We also obtain a lower bound in terms of minors in a symmetrized semiring

Stone algebras, conditional events, and three valued logic

Abstract: There are several equivalent ways to represent the set of conditional events, and in some the operations proposed by Goodman and Nguyen (1991) become much simpler, making the development of the theory much easier and much more concise. Such a development is carried out here using a representation whose relation to three-valued logic is analogous to that of Boolean algebras to two-valued logic, and in which the operations are simple and intuitive. There are many ways to extend the operations on events to operations on conditional events, but it is shown that there are only nine ways to extend intersection and nine ways to extend union so that the operations are Boolean polynomials of their arguments and are idempotent and commutative. Further, there is only one way to make such extensions so that the resulting structure is a bounded lattice extension of the space of events. These particular extensions turn out to be the operations proposed by Goodman and Nguyen. >

A Completeness Theorem fro Nondeterministic Kleene Algebras

Abstract: A generalization of Kleene Algebras (structures with +·*, 0 and 1 operators) is considered to take into account possible nondeterminism expressed by the + operator. It is shown that essentially the same complete axiomatization of Salomaa is obtained except for the elimination of the distribution P·(Q + R) = P·Q + P·R and the idempotence law P + P = P. The main result is that an algebra obtained from a suitable category of labelled trees plays the same role as the algebra of regular events. The algebraic semantics and the axiomatization are then extended by adding Ω and ∥ operator, and the whole set of laws is used as a touchstone for starting a discussion over the laws for deadlock, termination and divergence proposed for models of concurrent systems.

An approach to a dual of regular closure operators

Abstract: New techniques are introduced to obtain a completely symmetric analogue of the Salbany regular closure construction in a categorical setting. Galois connections and sink factorization structures are used as primary tools for the investigation. Factorizations of key Galois connections that represent such closure constructions are obtained as are detailed formulas for the construction of either canonical idempotent or weakly hereditary closure operators from arbitrary classes of objects and special classes of morphisms.

Braids and their monotone clones

Abstract: In this paper we investigate properties of the monotone clones of certain ordered sets known asbraids. This class of ordered sets arose naturally in the study of how the clone of monotone functions on an ordered set could satisfy, or fail to satisfy, Mal'cev conditions. One version of the main result can be stated as follows. IfB is a finite braid with reachr(B)>2 (defined in the text), then the only idempotent order-preserving functionsf∶Bn→B are then projections. It then follows, for example, that no algebra of monotone functions on a finite braidB withr(B)>2 generates a congruence-modular variety.

On modules for which the finite exchange property implies the countable exchange property

Abstract: It has been a long standing open problem whether the finite exchange property implies the full exchange property for an arbitrary module The main results of this paper are Theorem 11: For modules whose idempotent endomorphisms are central, the finite exchange property implies the countable exchange property, and Theorem 211: Over a ring with ace on essential right ideals, the finite exchange property implies the full exchange property for every quasi-continuous module The latter can be viewed as a partial affirmative answer to an open problem of Mohamed and Muller [8]

The set of idempotents of a completely regular semigroup as a binary algebra

Abstract: A semigroup S is called E-solid if and only if for all idempotents e, f, g ∈ S such that e L f R g there exists an idempotent h ∈ S such that e R h L g . Each completely regular semigroup is E-solid. We characterise the idempotents of an arbitrary E-solid regular semigroup as a set with a binary operation on it satisfying a given finite set of identities.

Construction of self-dual morphological operators and modifications of the median

Abstract: The median operator is a nonlinear (morphological) image transformation which has become very popular because it can suppress noise while preserving the edges It treats the foreground and background of an image in an identical way that is, it is a self-dual operator Unfortunately, the median operator lacks the idempotence property: it is not a morphological filter This paper gives a complete characterization of morphological operators on discrete binary images which are increasing, translation invariant, and self-dual Furthermore, it presents a general method for the modification of an increasing operator such that it becomes activity-extensive Such modifications lead to idempotent operators under iteration The general procedure is illustrated by giving several modifications of the 3/spl times/3 median operator >

Regular closure operators

Abstract: In an 〈E,M〉-categoryX for sinks, we identify necessary conditions for Galois connections from the power collection of the class of (composable pairs) of morphisms inM to factor through the “lattice” of all closure operators onM, and to factor through certain sublattices. This leads to the notion ofregular closure operator. As one byproduct of these results we not only arrive (in a novel way) at the Pumplun-Rohrl polarity between collections of morphisms and collections of objects in such a category, but obtain many factorizations of that polarity as well. (One of these factorizations constituted the main result of an earlier paper by the same authors). Another byproduct is the clarification of the Salbany construction (by means of relative dominions) of the largest idempotent closure operator that has a specified class ofX-objects as separated objects. The same relation that is used in Salbany's relative dominion construction induces classical regular closure operators as described above. Many other types of closure operators can be obtained by this technique; particular instances of this are the idempotent and modal closure operators that in a Grothendieck topos correspond to the Grothendieck topologies.