Showing papers on "Idempotence published in 1995"

Characterizing spatial quantales

Abstract: We will prove that an idempotent, right-sided quantale is spatial iff it is a subquantale of a product of simple quantales.

On a class of semigroup pseudovarieties without finite pseudoidentity basis

Abstract: We show that every semigroup pseudovariety containing a group not in the subpseudovariety generated by all idempotent generated members of has no finite basis of pseudoidentities provided the five-element idempotent generated 0-simple semigroup lies in . This gives, in particular, a counterexample to a conjecture by J. Almeida.

Rank problems for composite transformations

Abstract: Let (X, F) be a pair consisting of a finite set X and a set F of transformations of X, and, let and F(l) denote, respectively, the semigroup generated by F and the part of consisting of the transformations determined by a generator sequence of length no more than a given integer l. We show the following: • The problem whether or not, for a given pair (X, F) and a given integer r, there is an idempotent transformation of rank r in is PSPACE-complete. • For each fixed r≥1, it is decidable in a polynomial time, for a given pair (X, F), whether or not contains an idempotent transformation of rank r, and, if yes then a generator sequence of polynomial length composing to an idempotent transformation of rank r can be obtained in a polynomial time. • For each fixed r≥1, the problem whether or not, for a given (X, F) and l, there is an idempotent transformation of rank r in F(l) is NP-complete. • For each fixed r≥2, to decide, for a given (X, F), whether or not contains a transformation of rank r is NP-hard.

Fully Abstract Models for Nondeterministic Regular Expressions

Abstract: Regular expressions and Kleene Algebras have been a direct inspiration for many constructs and axiomatizations for concurrency models. These, however, put a different stress on nondeterminism. With concurrent interpretations in mind, we study the effect of removing the idempotence law X+X=X and distribution law X·(Y+Z)=X·Y +X·Z from Kleene Algebras. We propose an operational semantics that is sound and complete w.r.t. the new set of axioms and is fully abstract w.r.t. a denotational semantic based on trees. The operational semantics is based on labelled transition systems that keep track of the performed choices and on a preorder relation (we call it resource simulation) that takes also into account the number of states reachable via every action.An important property we exhibit is that resource bisimulation equivalence can be obtained as the kernel of resource simulation.

On the idempotence and stability of kernel functors

Abstract: A kernel functor (equivalently, a left exact torsion preradical) is a left exact subfunctor of the identity on the category R-mod of left R-modules over a ring R with identity. A kernel functor is said to be idempotent if, in addition, σ satisfies σ(M / σ(M)) = 0 for every M ∊ R-mod. To every kernel functor / there corresponds a unique topologizing filter ℒσ = {I Ⅰ σ (R/I) = R/I} of left ideals and a unique class ℱσ = {M ∊ R-mod Ⅰ σ(M) = M} that is closed under homomorphic images, submodules, and direct sums. The idempotence of σ is characterized by either of the following additional conditions:(1) if I ∊ ℒσ, K ⊆ I, and (K:x) = {r ∊ R ∣ rx ∊ K} ∊ ℒσ for each x ∊ I, then K ∊ ℒ or(2) ℱσ is closed under extensions of one member of ℱσ by another member of ℱσ Idempotent kernel functors are important since they are the tool used to construct localization functors. For M∊ R-mod, let E(M) denote the injective hull of M. A kernel functor σ is called stable if Mℱ implies that E(M) ∊ ℱσ For more information about kernel functors, see [6], [7], [14], and [15].

The joint of the pseudovarieties of idempotent semigroups and locally trivial semigroups

Abstract: This article solves a problem proposed by Almeida: the computation of the join of two well-known pseudovarieties of semigroups, namely the pseudovariety of bands and the pseudovariety of locally trivial semigroups. We use a method developed by Almeida, based on the theory of implicit operations.

Nearly prime subsemigroups ofβℕ

Abstract: We show that there exist relatively small subsemigroupsM ofβℕ with the property that ifp+q andq+p are inM then bothp andq are inM + ℤ. We also show that it is consistent with the usual axioms of set theory that there is some idempotent e inβℕ such that ifp+q=e, then bothp andq are ine + ℤ.

Orthomodular Posets of Idempotents in Finite Rings of Matrices

Abstract: The idempotents, resp. Hermitian idempotents, of a unital ring, resp. involutive unital ring, form an orthomodular poset. We study these Orthomodular posets for rings of matrices over the integers modulom or over Galois fields. In analogy to the Hilbert space situation we look for idempotent matrices (projections) corresponding to splitting subspaces of finite-dimensional vector spaces.

The geometry of finite dimensional pseudomodules

Abstract: A semimodule M over an idempotent semiring P is also idempotent. When P is linearly ordered and conditionally complete, we call it a pseudoring, and we say that M is a pseudomodule over P. The classification problem of the isomorphy classes of pseudomodules is a combinatorial problem which, in part, is related to the classification of isomorphy classes of semilattices. We define the structural semilattice of a pseudomodule, which is then used to introduce the concept of torsion. Then we show that every finitely generated pseudomodule may be canonically decomposed into the "sum" of a torsion free subpseudomodule, and another one which contains all the elements responsible for the torsion of M. This decomposition is similar to the classical decomposition of a module over an integral domain into a free part, and a torsion part. It allows for a great simplification of the classification problem, since each part can be studied separately. In sub-pseudomodules of the free pseudomodule over m generators, the torsion free part, also called semiboolean, is completely characterized by a weighted oriented graph whose set of vertices is the structural semilattice of M. Partial results on the classification of the isomorphy class of a torsion sub-pseudomodule of P/sup m/ with m generators will also be presented.

Doubly stochastic matrices whose squares are idempotent

Abstract: Formulas for the doubly stochastic square roots of the idempotents are presented. Such square roots must necessarily satisfy per A2 ≤perA.

The allowance of idempotent of sign pattern matrices

Abstract: A matrix whose entries consist of the symbols +, - and 0 is called a sign pattern matrix. In [1], a graph theoretic characterization of sign idempotent pattern matrices was given. A question was given for the sign patterns which allow idempotence. We characterized the sign patterns which allow idempotence in the sign idempotent pattern matrices.

Every nearly idempotent plain algebra generates a minimal variety

Abstract: An algebra A is plain if it is finite, simple and has no non-trivial proper subalgebras. An element 0 ∈ A is an idempotent element if {0} is a subuniverse and is a non-idempotent element otherwise. A is idempotent if each of its elements is idempotent. In this paper we shall say that A is nearly idempotent if A has at least one idempotent element and Aut(A) acts transitively on the non-idempotent elements. In [2], Ágnes Szendrei proves that every idempotent plain algebra generates a minimal variety by showing that an idempotent plain algebra with more than two elements generates a congruence modular variety. The proof is not long, but it relies on the classification theorem in [1] for idempotent plain algebras of size > 2. The proof in [1] of this classification theorem covers several pages. The argument in [2] is completed by directly examining the congruence modular case and the 2-element case and proving for both that an idempotent plain algebra generates a minimal variety. Here we give a short proof of the result using only “V = HSP”. With Theorem 4 we show how to boost the result to a proof that every nearly idempotent plain algebra generates a minimal variety. We say that V satisfies condition (E) if V has a unary term e such that for all basic operations f the identity f(e(x), . . . , e(x)) = e(x) holds. If A is an idempotent plain algebra, then V = V(A) satisfies condition (E) with e(x) = x. If A is plain and V(A) is not minimal, then there is a plain algebra B ∈ V(A) which generates a minimal subvariety. Clearly, A 6∼= B in this case. Szendrei’s result can be deduced from the following lemma, since it shows that when A is plain and idempotent and B ∈ V(A) is plain (and of course idempotent), then A ∼= B. LEMMA 1 If A is plain, V = V(A) satisfies condition (E) and B ∈ V is idempotent and plain, then A ∼= B. Proof: Assuming the hypotheses of the lemma we can find m, a subalgebra C ≤ Am and a congruence θ on C such that C/θ ∼= B. Among all such situations, choose one so that |C| is minimal. If η is a projection kernel restricted to C and η ≤ θ, then B ∈ H(C/η) = HS(A). A is plain and B is nontrivial, so this yields A ∼= B and finishes the proof. Otherwise, for each projection kernel η there is a pair (a, b) ∈ η − θ. We claim that (e(a), e(b)) ∈ η − θ as well. Of course, (a, b) ∈ η implies (e(a), e(b)) ∈ η. Since C/θ ∼= B is idempotent, e(x) θ x holds on C. Hence e(a) θ a and e(b) θ b hold. Now (a, b) 6∈ θ implies (e(a), e(b)) 6∈ θ by transitivity. ∗Supported by a fellowship from the Alexander von Humboldt Stiftung.

Sur les algèbres de Moufang commutatives

Abstract: The structure of commutative Moufang algebras is studied. We show in particular that the set of idempotent elements is contained in the center of the algebra thus it has a boolean ring structure. Simple and semi-simple commutative Moufang algebras are associative and the associator (x, y, z) is always in the nilradical.