Showing papers on "Idempotence published in 1998"
01 Jan 1998
TL;DR: It is shown that by choosing suitable fuzzy logical operators, all classical duality and other relationships can be preserved and it is possible to assure the idempotence of the fuzzy closing and fuzzy opening.
Abstract: A logical approach to the fuzzification of binary (mathematical) morphology is presented The fuzzy dilation and fuzzy erosion are introduced independently, and are based on the fuzzy logical operators’ conjunctor’ and’ implicator’ In this way, duality relationships are not forced from the very beginning It is shown that by choosing suitable fuzzy logical operators, all classical duality and other relationships can be preserved Following a similar line of reasoning, it is possible to assure the idempotence of the fuzzy closing and fuzzy opening
110 citations
TL;DR: In this article, the concept of hyperidentity was introduced for binary representations and the concepts of coidentity, homomorphisms and automorphisms of algebras of the same arithmetic type.
Abstract: ContentsIntroduction ??1. The concept of hyperidentity. Hyperidentities of binary representations1.1. The concept of hyperidentity: examples1.2. Binary Cayley theorems for semigroups, idempotent and commutative semigroups, groups and the multiplicative groups of fields1.3. Stochastic algebras. The concept of coidentity ??2. The category of algebras of the same arithmetic type2.1. Homomorphisms and automorphisms of algebras of the same arithmetic type2.2. Hypervarieties of algebras ??3. The category of systems of the same arithmetic type. Nonclassical semantics in second order language ??4. Non-trivial associative and distributive hyperidentities in algebras that are invertible or close to being invertible Bibliography
62 citations
TL;DR: It is derived that if A is an abelian algebra and (A) satisfies an idempotent Mal'cev condition which fails to hold in the variety of semilattices, then A is affine, and it is refined by showing that abelIAN algebras are quasi-affine in such varieties.
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 (A) satisfies an idempotent Mal'cev condition which fails to hold in the variety of semilattices, then A is affine.
61 citations
Journal Article•
TL;DR: In this paper, an algebraic approach to idempotent functional analysis is presented, where the basic concepts and results of linear functional analysis are expressed in purely algebraic terms.
Abstract: In this paper, we present an algebraic approach to idempotent functional analysis, which is an abstract version of idempotent analysis. The basic concepts and results are expressed in purely algebraic terms. We consider idempotent versions of certain basic results of linear functional analysis, including the theorem on the general form of a linear functional and the Hahn-Banach and Riesz-Fischer theorems.
30 citations
TL;DR: For a given idempotent p and some element σ from a differential associative ring, the authors introduced a gauge transformation λp + σ with the spectral parameter λ that leaves some linear operators form invariant.
Abstract: For a given idempotent p and some element σ from a differential associative ring, we introduce gauge transformation λp + σ with the spectral parameter λ that leaves some linear operators form invariant. The explicit form of the σ is derived for the generalized Zakharov-Shabat problem. The maps that factorize Darboux transformations are referred to as elementary ones. Binary transformations that correspond to the iteration of elementary maps with the special choice of solutions of the direct and conjugate problems are introduced and widen the potential reductions set. We show an infinitesimal version of iterated transforms. The spectral operator and soliton theories applications are outlined. The nonabelian N-wave interaction equation modification with additional linear terms, its zero curvature representation, soliton solutions, and their stability with respect to infinitesimal deformations are studied.
22 citations
14 Sep 1998
TL;DR: This paper provides a generalisation of the abstraction function for Sharing that can be applied to any language, with or without the occur-check, and the results for safeness, idempotence and commutativity for abstract unification using this abstraction function are given.
Abstract: It is important that practical data flow analysers are backed by reliably proven theoretical results. Abstract interpretation provides a sound mathematical framework and necessary generic properties for an abstract domain to be well-defined and sound with respect to the concrete semantics. In logic programming, the abstract domain Sharing is a standard choice for sharing analysis for both practical work and further theoretical study. In spite of this, we found that there were no satisfactory proofs for the key properties of commutativity and idempotence that are essential for Sharing to be well-defined and that published statements of the safeness property assumed the occur-check. This paper provides a generalisation of the abstraction function for Sharing that can be applied to any language, with or without the occur-check. The results for safeness, idempotence and commutativity for abstract unification using this abstraction function are given.
17 citations
TL;DR: A discussion of sheaves and presheaves over a right sided idempotent quantale in a fashion that is similar to the way that these objects are conceivedover complete Heyting algebras by Fourman and Scott in 5].
Abstract: We present a discussion of sheaves and presheaves over a right sided idempotent quantale in a fashion that is similar to the way that these objects are conceivedover complete Heyting algebras by Fourman and Scott in 5]. The idea of a quantale originated with C.J.Mulvey ((6]) as an attempt to code a lattice theoretic construct that might be appropriate to obtain, for non commutative C algebras, an analogue of the classical duality between commutative C algebras and compact Hausdorr spaces. We cite 8] as a general reference on the circle of ideas connected with quantales. A number of authors have studied the possibility of extending the notions of sheaf and presheaf over a complete Heyting algebra (cHa) or frame to this new context ((7], 2], 1], 4]). It should be mentioned that an exposition of the content of 2] can be found in 8]. We consider 5] as a basic reference for presheaves and sheaves over complete Heyt-ing algebras. We discuss here how much of the theory in that paper can be carried over to the quantale setting. We shall focus our attention on quantales which are referred to as idempotent and right sided (deenitions will follow). We consider as central the notion of extensionality or separation, in the sense that`sections that coincide locally are equal'. Our treatment, although related to that in the references mentioned above, is distinct in a number of aspects, such as the concepts of presheaf, morphisms, completion and characteristic maps. We also prepare a discussion of sheaves and presheaves over two sided quantales, to appear separately. In Section 1 we recall the deenition of quantale and some of the basic properties of this concept. The notion of Q-set is studied in Section 2, where we also introduce the concept of extensionality. Presheaves are the main theme of Section 3, while Section 4 discusses the category of sheaves over a right sided quantale. Section 5 is 545 L. 546 Sheaves over Right Sided Idempotent Quantales devoted to the main properties of characteristic functions, a convenient way to study the structure of subsheaves. In Section 6 we compute the characteristic functions of the quantiiers 8 and 9 relative to the theory of sheaves set down in Section 4. We would like to express our thanks to F.Borceux and C.J.Mulvey, who kindly sent us copies of 2], 1], 4] and 7], respectively.
13 citations
01 Jan 1998
TL;DR: In this paper it was shown that under some finiteness conditions in a (not necessarily commutative and not necessarily cancellative) semigroup every non-unit is a product of weakly irreducible elements.
Abstract: In this paper we prove that under some finiteness conditions in a (not necessarily commutative and not necessarily cancellative) semigroup every non-unit is a product of weakly irreducible elements. In commutative, finitely generated semigroups every infinitely divisible element is idempotent. Without commutativity this is not true. An interesting open problem is to find necessary and sufficient conditions for this implication.
8 citations
TL;DR: The theory of idempotence of operators of the form ɛ∨id∧δ on a modular lattice ℒ can be applied to the design of Idempotent morphological filters removing isolated spots in digital pictures.
Abstract: We study the idempotence of operators of the form ɛ∨id∧δ (where ɛ≤δ and both ɛ and δ are increasing) on a modular lattice ℒ, in relation to the idempotence of the operators ɛ∨id and id∧δ. We consider also the conditions under which ɛ∨id∧δ is the composition of ɛ∨id and id∧δ. The case where δ is a dilation and ɛ an erosion is of special interest. When ℒ is a complete lattice on which Minkowski operations can be defined, we obtain very precise conditions for the idempotence of these operators. Here id∧δ is called an annular opening, ɛ∨id is called an annular closing, and ɛ∨id∧δ is called an annular filter. Our theory can be applied to the design of idempotent morphological filters removing isolated spots in digital pictures.
8 citations
TL;DR: The role of idempotent elements and atoms in the theory of ideal and filter in D-posets is studied in this paper, where the authors define the notion of ideal elements and filters.
Abstract: Ideals and filters in D-posets are defined. Therole of idempotent elements and atoms in the theory ofideals and filters is studied.
7 citations
TL;DR: In this article, it was shown that ν( R (m) = log 2 m ⌉ for every m ǫ ≥ 2, where mǫ denotes the direct sum of m copies of R. The latter result corrects an error by Krupnik (Comm. Algebra 20, 1992, 3251-3257).
TL;DR: In this article, the Cayley theorem for distributive lattices was shown to be generalized to non-distributive lattice or quasilattices without changing the definitions of ∨ and ∧.
Abstract: In this paper we define a lattice order on a set F of binary functions. We then provide necessary and sufficient conditions for the resulting algebra \(\mathfrak{L}\)F to be a distributive lattice or a Boolean algebra. We also prove a ‘Cayley theorem’ for distributive lattices by showing that for every distributive lattice \(\mathfrak{L}\), there is an algebra \(\mathfrak{L}\)F of binary functions, such that \(\mathfrak{L}\) is isomorphic to\(\mathfrak{L}\)F and we show that \(\mathfrak{L}\)F is a distributive lattice iff the operations ∨ and ∧ are idempotent and cummutative, showing that this result cannot be generalized to non-distributive lattices or quasilattices without changing the definitions of ∨ and ∧. We also examine the equational properties of an Algebra \(\mathfrak{U}\) for which \(\mathfrak{L}_\mathfrak{U}\), now defined on the set of binary \(\mathfrak{U}\)-polynomials is a lattice or Boolean algebra.
TL;DR: It is proved that the class F 2 ( D ) of all idempotent fuzzy subsets of D, where D is a semigroup in which the cancellation laws are valid, forms a complete lattice under the pointwise definition of order.
01 Jan 1998
TL;DR: In this article, the radical theory of algebras with B-action was initiated, where B is a fixed Boolean ring and the ideal of B is defined in terms of ideals of B. In two special cases (universal classes of \(\omega \)-groups with b-action and idempotent classes with Baction), these ideal-defined classes are sublattices of the lattice of radicals, and characterise semisimplicity in such cases.
Abstract: We initiate the radical theory of algebras with B-action where B is a fixed Boolean ring. We consider lattices of classes of algebras defined in terms of ideals of B. In two special cases (universal classes of \(\omega \)-groups with B-action and idempotent algebras with B-action), these ideal-defined classes are sublattices of the lattice of radicals, and we characterise semisimplicity in such cases.
01 Jan 1998
TL;DR: This work develops some interesting properties related to their structure as well as give several necessary conditions for idempotent functions with two minimal primes and pave the way toward a characterization of the structure of this class of functions.
Abstract: Monotone Boolean functions have been extensively studied in the area of nonlinear digital filtering, specifically stack and morphological filtering. In fact, any Stack Filter of window-width n is uniquely specified by a monotone Boolean function of n variables. Similarly, the Stacking Property obeyed by all stack filters is the monotonicity of these Boolean functions [5]. In this paper, we focus on idempotent monotone Boolean functions and develop some interesting properties related to their structure as well as give several necessary conditions for idempotent functions with two minimal primes. The idempotence property implies that a root signal is obtained in one pass. That is, subsequent filter passes do not alter the signal. By developing some structural properties of these functions, we pave the way toward a characterization of the structure of this class of functions. Such a characterization would prove to be most useful in the theory of optimal stack filtering and would facilitate the search for optimal idempotent stack filters. To conserve space, we will omit the proofs of some propositions and lemmas which are relatively straightforward to construct.
TL;DR: In this article, Huang et al. discuss the Green's classes, idempotent elements, maximal subgroups, and regular elements in GMn(R) with the sandwich semigroup of GMn with R. If G is an n-order cyclic group, their results are exactly the results of Wenchao Huang in [Linear Algebra Appl., 25 (1979), pp 135--160].
Abstract: Let GMn be the semigroup of all the n x n (n \geq 2)$ group Boolean matrices, and let R be a nonzero element in GMn, where G is an n-order Abelian group. The sandwich semigroup of GMn with the sandwich element R is denoted by GMn(R). The purpose of this paper is to discuss the Green's classes, idempotent elements, maximal subgroups, and regular elements in GMn(R). If G is an n-order cyclic group, our results are exactly the results of Wenchao Huang in [Linear Algebra Appl., 25 (1979), pp. 135--160].
TL;DR: It is proved that an irreducible building of spherical type and of rank at least 2 has the projection property and holds not only for the case of a product of two copies of X but for any finite number of copies ofX and is thus similar to Arrow's theorem.
Proceedings Article•
01 Sep 1998TL;DR: An algorithm for testing a given monotone Boolean function for idempotence is presented and the set ofidempotent monot one Boolean functions of 5 variables is provided.
Abstract: This paper focuses on the class of idempotent monotone Boolean functions. Monotone Boolean functions correspond to an important class of non-linear digital filters called Stack Filters. The idempotence property implies that applying such a filter produces a root signal in one pass. We present an algorithm for testing a given monotone Boolean function for idempotence and provide the set of idempotent monotone Boolean functions of 5 variables.
TL;DR: Fully idempotent near-rings are defined and characterized in this article, which yields information on the lattice of ideals of fully idempotsent rings and near-ring.
Abstract: Fully idempotent near-rings are defined and characterized which yields information on
the lattice of ideals of fully idempotent rings and near-rings. The space of prime ideals is topologized and
a sheaf representation is given for a class of fully idempotent near-rings which includes strongly regular
near-rings.
Proceedings Article•
01 Jan 1998TL;DR: In this paper, a generalisation of the abstraction function for Sharing that can be applied to any language, with or without the occur-check, is presented, and the results for safeness, idempotence and commutativity for abstract unification using this abstraction function are given.
Abstract: It is important that practical data flow analysers are backed by reliably proven theoretical results Abstract interpretation provides a sound mathematical framework and necessary generic properties for an abstract domain to be well-defined and sound with respect to the concrete semantics In logic programming, the abstract domain Sharing is a standard choice for sharing analysis for both practical work and further theoretical study In spite of this, we found that there were no satisfactory proofs for the key properties of commutativity and idempotence that are essential for Sharing to be well-defined and that published statements of the safeness property assumed the occur-check This paper provides a generalisation of the abstraction function for Sharing that can be applied to any language, with or without the occur-check The results for safeness, idempotence and commutativity for abstract unification using this abstraction function are given
Posted Content•
TL;DR: In this paper, the role of idempotence in the known axiomizations of location functionals is explained, and the distribution of ideme-potent and sufficient statistics is derived for parametric families of location.
Abstract: Idempotence is a well-known property of functionals of location. It means that the value of the functional at a singular distribution must be identically to the mass point of this distribution. First, we explain the role of idempotence in the known axiomizations of location functionals. Then we derive the distribution of idempotent and sufficient statistics. In the special cases of parametric families of location we get the so-called power-n-distributions. Power-n-distributions again are distributions with a parameter of location and can be derived from every location family for which the density is constrained. Additionally we show that the completeness of the populations family insures the completeness of the family of power-n-distributions. And at last, we give a further, now very easy proof that the normal distribution is the only one for which a idempotent, sufficient and unbiased estimator attains the Cramer-Rao-lower bound.
Journal Article•
TL;DR: In this article, it was shown that the same is essentially true even if we do not assume existence of an identity, and the authors formulated the main results in such a way that the result of [5] is an easy consequence, following as a special case.
Abstract: In the recent paper [5] we showed that certain algebras, with identity, can be represented as algebras of continuous functions on a Boolean space (a totally disconnected compact space). Now we shall show that the same is essentially true even if we do not assume existence of an identity. More specifically, we shall characterize the space Cp(S) of continuous complexvalued functions on a Boolean space S, each vanishing on certain point pES, specified in advance. We shall formulate the main results in such a way that the result of [5] is an easy consequence, following as a special case. Our proofs will use a different approach (we shall utilize Gelfand theory rather than that of Stone).
01 Jan 1998
TL;DR: Fully idempotent near-rings are defined and characterized in this paper, which yields information on the lattice of ideals of fully idempotsent rings and near-ring.
Abstract: Fully idempotent near-rings are defined and characterized which yields information on the lattice of ideals of fully idempotent rings and near-rings. The space of prime ideals is topologized and a sheaf representation is given for a class of fully idempotent near-rings which includes strongly regular