Showing papers on "Idempotence published in 1996"
TL;DR: Normalized rewriting is introduced, a new rewrite relation which generalizes former notions of rewriting modulo a set of equations E , dropping some conditions on E , and gives a new completion algorithm for normalized rewriting.
85 citations
TL;DR: The general form of an idempotent, associative, nondecreasing and continuous binary aggregation operation in a connected order topological space is given.
Abstract: We give the general form of an idempotent, associative, nondecreasing and continuous binary aggregation operation in a connected order topological space. The particular case of the unit interval is studied and the choice of weights is also analized. Possible generalizations for more than two arguments are also proposed.
49 citations
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TL;DR: In this paper, it was shown that the linear commutator is definable in terms of the centralizer relation in any variety satisfying a nontrivial idempotent Mal'cev condition.
Abstract: We clarify the relationship between the linear commutator and the ordinary commutator by showing that in any variety satisfying a nontrivial idempotent Mal'cev condition the linear commutator is definable in terms of the centralizer relation. We derive from this that abelian algebras are quasi-affine in such varieties. We refine this by showing that if A is an abelian algebra and V(A) satifies an idempotent Mal'cev condition which fails to hold in the variety of semilattices, then A is affine.
48 citations
TL;DR: Decompositions formulas of open-overcondensations in terms of such operators are given; they parallel the well-known decomposition formulas for openings.
45 citations
27 Mar 1996
TL;DR: An equational proof, using Kleene algebra with tests and commutativity conditions, of the following classical result: every while program can be simulated by a while program with at most one while loop.
Abstract: We give an equational proof, using Kleene algebra with tests and commutativity conditions, of the following classical result: every while program can be simulated by a while program with at most one while loop. The proof illustrates the use of Kleene algebra with extra conditions in program equivalence proofs. We also show, using a construction of Cohen, that the universal Horn theory of *-continuous Kleene algebras is not finitely axiomatizable.
44 citations
27 Jul 1996
TL;DR: The problem of unifying these two algorithms for building convergent rewrite systems from a given equational axiomatization into a common general one arised and was described by Jouannand and March~ in 1990.
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.
42 citations
01 Jan 1996
TL;DR: Semirings are those that originate from rings, by cancelling the assumption that a ring-like universal algebra has to be a group as discussed by the authors, and semirings with different properties have become important in theoretical computer science.
Abstract: This chapter discusses semirings and semifields. Among many concepts of universal algebras, semirings are those that originate from rings, by cancelling the assumption that ( R , +) has to be a group. Depending on how much other ring-like properties are also cancelled or added, various different concepts of semirings (S, +, ·) have been considered. Semirings with different properties have become important in theoretical computer science. A universal algebra S = (S, +, ·) with a nonempty set S and two binary operations, written as addition and multiplication, is called a “semiring” if (S, +) and (S,·) are arbitrary semigroups, such that a ( b + c ) = ab + ac and ( b + c ) a = ba + ca hold for all a, b, c ∈ S . In particular, ( S , +,·) is called a “proper semiring” if ( S , +) is not a group. Each distributive lattice is a semiring, clearly commutative and idempotent with respect to both operations. It has a zero or an identity if it is bounded from below or above, respectively.
36 citations
TL;DR: In this paper, a simple example is given to illustrate that an idempotent state may not be the Haar state of any subgroup in the case of compact quantum groups.
Abstract: A simple example is given to illustrate that an idempotent state may not be the Haar state of any subgroup in the case of compact quantum groups.
31 citations
TL;DR: In this article, the authors provide a possible tool to study the invertibility of local cosets which arise after localization of convolution type operators, where the basic observation is that the local algebras in some cases are generated by a finite number of elements p which are idempotent in the sense that p 2=p.
22 citations
TL;DR: In this article, it was shown that the semigroup A 2 is generated by transposition, similarity, and the operators X → X τ for fixed arbitrary injective endomorphisms τ on F, and X → σ (tr X ) for fixed additive maps σ from F to M n (F ) with σ(1) = O.
21 citations
TL;DR: In this article, the existence of an idempotent element in train algebras of rank greater than 3 is investigated and the conditions that ensure such an element exist are established.
Abstract: The existence of idempotent elements in train algebras of rank greater than 3 is an open question to be solved. Recent H. Guzzo results [7] on train algebras of rank 4 are based on the underlying assumption of the existence of an idempotent. In the present paper we establish the conditions that ensure the existence of such an idempotent. We also give additional properties on the Peirce decomposition which allow us to characterize some train algebras of rank 4. Finally, we give a characterization of the train algebras of rank 4 which are power-associative algebras or Jordan algebras.
TL;DR: In this paper, the hyperpower method is generalized by insizing an idempotent matrix P. The convergence behavior of Bqk is analyzed and the results are applied to give a detailed investigation of this iteration.
TL;DR: In this paper, necessary and sufficient conditions are presented for a square matrix over an arbitrary field to be a product of k ≥ 1 idempotent matrices of prescribed nullities.
Abstract: Necessary and sufficient conditions are presented for a square matrix over an arbitrary field to be a product of k ≥ 1 idempotent matrices of prescribed nullities.
TL;DR: In this article, a non-zero idempotent element can be represented as a sum of two nilpotent elements, where the latter is the sum of a ring and the former is a ring.
Abstract: Abstract In this paper we study in which rings a non-zero idempotent element can be presented as a sum of two nilpotent elements.
TL;DR: In this article, it was shown that every additively-idempotent semiring can be embedded in a finitary complete semiring, and that the classical identities of Kleene semirings over idempotent semiirings are independent.
Abstract: We prove that every additively-idempotent semiring can be embedded in a finitary complete semiring. From this we obtain, among other results, that the classical identities of Kleene semirings over idempotent semirings are independent.
TL;DR: The intuitionistic tableau procedure is generalized and it is proved that this generalized tableau method is sound for the semantics of residuated propositional calculus (RPC) and whenever a formula 0 is tableau provable, it is deducible in RPC.
TL;DR: In this paper, mutually orthogonal idempotent Latin squares of orders 22 and 26 are constructed, which can be used to obtain 3 HMOLS of type 522 and type 2322 and to obtain a (110, 5, 1)-PMD and a (130 5, 2)-HMOLS.
TL;DR: The call-by-value CPS transformation is rephrase to make it syntactically idempotent, modulo eta-reduction of the newly introduced continuation.
Abstract: The CPS (continuation-passing style) transformation on typed lambda-terms has an interpretation in many areas of Computer Science, such as programming languages and type theory. Programming intuition suggests that in effect, it is idempotent, but this does not directly hold for all existing CPS transformations (Plotkin, Reynolds, Fischer, etc.). We rephrase the call-by-value CPS transformation to make it syntactically idempotent, modulo eta-reduction of the newly introduced continuation. Type-wise, iterating the transformation corresponds to refining the polymorphic domain of answers.
01 Jan 1996
TL;DR: In this paper, the authors show how to accelerate simple cases of strictness analysis for first-order functional programs and, perhaps more successfully, groundness analysis of logic programs on distributive lattices.
Abstract: A theorem by Schroder says that for a certain natural class of functions F: B → B defined on a Boolean lattice B, F(x)=F(F(F(x))) for all x ∃ B. An immediate corollary is that if such a function is monotonic then it is also idempotent, that is, F(x)=F(F(x)). We show how this corollary can be extended to recognize cases where recursive definitions can immediately be replaced by an equivalent closed form, that is, they can be solved without Kleene iteration. Our result applies more generally to distributive lattices. It has applications for example in the abstract interpretation of declarative programs and deductive databases. We exemplify this by showing how to accelerate simple cases of strictness analysis for first-order functional programs and, perhaps more successfully, groundness analysis for logic programs.
TL;DR: For appropriate k, the lattice of subvarieties of the variety of all idempotent algebras of the completely regular semigroups over groups that belong to is of the power of the continuum.
Abstract: Let be the semigroup variety determined by the identity xm=xm+k. For we define operations on the set E(S) of idempotents of S and thus obtain the idempotent algebra of S. For any subvariety of the idempotent algebras of the members of form a variety and yields a complete homomorphism of the lattice of subvarieties of onto the lattice of subvarieties of . The lattice contains a ∩-semilattice isomorphic to the ∩-semilattice of group varieties of exponent dividing k for every m≥1. In particular, for appropriate k, the lattice of subvarieties of the variety of all idempotent algebras of the completely regular semigroups over groups that belong to is of the power of the continuum. For any , ρ→ρ|E(S) yields a complete homomorphism of the congruence lattice of S into the lattice of equivalence relations on E(S).
18 Dec 1996
TL;DR: This work shows how to accelerate simple cases of strictness analysis for first-order functional programs and, perhaps more successfully, grounds analysis for logic programs by showing how to solve cases where recursive definitions can be replaced by an equivalent closed form.
Abstract: A theorem by Schroder says that for a certain natural class of functions F: B → B defined on a Boolean lattice B, F(x)=F(F(F(x))) for all x ∃ B. An immediate corollary is that if such a function is monotonic then it is also idempotent, that is, F(x)=F(F(x)). We show how this corollary can be extended to recognize cases where recursive definitions can immediately be replaced by an equivalent closed form, that is, they can be solved without Kleene iteration. Our result applies more generally to distributive lattices. It has applications for example in the abstract interpretation of declarative programs and deductive databases. We exemplify this by showing how to accelerate simple cases of strictness analysis for first-order functional programs and, perhaps more successfully, groundness analysis for logic programs.
TL;DR: In this article, the authors characterize the matrices A ∈ Cn(R) that belong to IR in the sense that A ∗ k = I R for some nonnegative integer k.