Idempotency Research Articles

A construction of an idempotent half-completion for quasi-uniform spaces

Császár (1963) and Deák (1991) have introduced the notion of half-completeness in quasi-uniform spaces which generalizes the well known notion of bicompleteness. In this paper, for any quasi-uniform space, we construct a half-completion, called standard half-completion. The standard half-completion coincides with the usual uniform completion in the case of uniform spaces. It is also an idempotent operation in the sense that the standard half-completion of a half-complete quasi-uniform space coincides (up to a quasi-isomorphism) with the space itself.

Smarandache Idempotents in Certain Types of Group Rings

In this paper we study S-idempotents of the group ring Z2G where G is a finite cyclic group of
 order . We give a condition on such that every nonzero idempotent element of the group ring
 Z2G is Smarandache idempotent and we find Smarandache idempotents of the group ring KG
 where K is an algebraically closed field of characteristic 0 and is a finite cyclic group.

Varieties with definable factor congruences

We study direct product representations of algebras in varieties. We collect several conditions expressing that these representations are definable in a first-order-logic sense, among them the concept of Definable Factor Congruences (DFC). The main results are that DFC is a Mal'cev property and that it is equivalent to all other conditions formulated; in particular we prove that V has DFC if and only if V has 0 & 1 and Boolean Factor Congruences. We also obtain an explicit first-order definition Φ of the kernel of the canonical projections via the terms associated to the Mal'cev condition for DFC, in such a manner that it is preserved by taking direct products and direct factors. The main tool is the use of central elements, which are a generalization of both central idempotent elements in rings with identity and neutral complemented elements in a bounded lattice.

Characterizations of derivations on triangular rings: Additive maps derivable at idempotents

Let T be a triangular ring. An element Z ∈ T is said to be a full-derivable point of T if every additive map δ from T into itself derivable at Z (i.e. δ ( A ) B + A δ ( B ) = δ ( Z ) for every A , B ∈ T with AB = Z ) is in fact a derivation. In this paper, under some mild conditions on triangular ring T , we show that some idempotent elements of T are full-derivable points. As an application, we get that, for any nontrivial nest N in a factor von Neumann algebra R , every nonzero idempotent element Q satisfying PQ = Q , QP = P for some projection P ∈ N is a full-derivable point of the nest subalgebra Alg N of R .

Semigroup of matrices acting on the max-plus projective space

We investigate the action of semigroups of d × d matrices with entries in the max-plus semifield on the max-plus projective space. Recall that semigroups generated by one element with projectively bounded image are projectively finite and thus contain idempotent elements. In terms of orbits, our main result states that the image of a minimal orbit by an idempotent element of the semigroup with minimal rank has at most d ! elements. Moreover, each idempotent element with minimal rank maps at least one orbit onto a singleton. This allows us to deduce the central limit theorem for stochastic recurrent sequences driven by independent random matrices that take countably many values, as soon as the semigroup generated by the values contains an element with projectively bounded image.

SemQA: SPARQL with Idempotent Disjunction.

The SPARQL LeftJoin abstract operator is not distributive over Union; this limits the algebraic manipulation of graph patterns, which in turn restricts the ability to create query plans for distributed processing or query optimization. In this paper, we present semQA, an algebraic extension for the SPARQL query language for RDF, which overcomes this issue by transforming graph patterns through the use of an idempotent disjunction operator Or as a substitute for Union. This permits the application of a set of equivalences that transform a query into distinct forms. We further present an algorithm to derive the solution set of the original query from the solution set of a query where Union has been substituted by Or. We also analyze the combined complexity of SPARQL, proving it to be NP-complete. It is also shown that the SPARQL query language is not, in the general case, fixed-parameter tractable. Experimental results are presented to validate the query evaluation methodology presented in this paper against the SPARQL standard to corroborate the complexity analysis and to illustrate the gains in processing cost reduction that can be obtained through the application of semQA.

On the structure of left and right F-, SM-, and E-quasigroups

It is proved that any left F-quasigroup is isomorphic to the direct product of a left F-quasigroup with a unique idempotent element and isotope of a special form of a left distributive quasigroup. The similar theorems are proved for right F-quasigroups, left and right SM- and E-quasigroups. Information on simple quasigroups from these quasigroup classes is given, for example, finite simple F-quasigroup is a simple group or a simple medial quasigroup. It is proved that any left F-quasigroup is isotopic to the direct product of a group and a left S-loop. Some properties of loop isotopes of F-quasigroups (including M-loops) are pointed out. A left special loop is an isotope of a left F-quasigroup if and only if this loop is isomorphic the direct product of a group and a left S-loop (this is an answer to Belousov 1a, problem). Any left FESM-quasigroup is isotopic to the direct product of an abelian group and a left S-loop (this is an answer to Kinyon-Phillips 2.8(2) problem). New proofs of some known results on the structure of commutative Moufang loops are presented.

SEMIGROUPS OF TREE LANGUAGES

Sets of terms of type τ are called tree languages (see [6]). There are several possibilities to define superposition operations on sets of tree languages. On the basis of such superposition operations we define binary associative operations on tree languages and investigate the properties of the arising semigroups. We characterize idempotent and regular elements and Green's relations [Formula: see text] and [Formula: see text]. Moreover, we determine constant, left-zero and right-zero subsemigroups and rectangular bands.

Applying Universal Algebra to Lambda Calculus

The aim of this article is double. From one side we survey the knowledge we have acquired these last ten years about the lattice of all λ-theories (equational extensions of untyped λ-calculus) and the models of lambda calculus via universal algebra. This includes positive or negative answers to several questions raised in these years as well as several independent results, the state of the art about the long-standing open questions concerning the representability of λ-theories as theories of models, and 26 open problems. On the other side, against the common belief, we show that lambda calculus and combinatory logic satisfy interesting algebraic properties. In fact the Stone representation theorem for Boolean algebras can be generalized to combinatory algebras and λ-abstraction algebras. In every combinatory and λ-abstraction algebra, there is a Boolean algebra of central elements (playing the role of idempotent elements in rings). Central elements are used to represent any combinatory and λ-abstraction algebra as a weak Boolean product of directly indecomposable algebras (i.e. algebras that cannot be decomposed as the Cartesian product of two other non-trivial algebras). Central elements are also used to provide applications of the representation theorem to lambda calculus. We show that the indecomposable semantics (i.e. the semantics of lambda calculus given in terms of models of lambda calculus, which are directly indecomposable as combinatory algebras) includes the continuous, stable and strongly stable semantics, and the term models of all semisensible λ-theories. In one of the main results of the article we show that the indecomposable semantics is equationally incomplete, and this incompleteness is as wide as possible.

Idempotents and regular elements of complete semigroups of binary relations

As is known (see [5, 6]), idempotent elements of semigroups play an important role in the investigation of semigroups themselves. Moreover, it is known that any semigroup can be isomorphically embedded into some semigroup of binary relations on some nonempty set X. This has generated great interest in the investigation of idempotent binary relations. Different authors used different approaches to study the construction of these elements and obtained different answers to the question under consideration (see, e.g., [1–3, 7, 8]). In this work, first, we characterize the complete X-semilattices of unions D for which there exist idempotent binary relations such that the set of all their cuts to the elements of D coincides with the given semilattice. Then we give a description of the structure of these idempotent binary relations in the language of limiting elements of some subset of the semilattice D (see Lemma 1 and Theorem 5) and show how we can find all idempotents of the complete group of binary relations BX (D) defined by the complete X-semilattice of unions D (see Theorem 6 and Corollary 4). Moreover, with the help of chains of the semilattice D, we give a rule for constructing the semigroups of idempotent elements of the semigroup BX(D) (see Theorem 7). In Theorem 9, we give a description of the structure of all regular elements of the semigroup BX (D).

Consistency properties and set based logic programming

Blair et al. (2001) developed an extension of logic programming called set based logic programming. In the theory of set based logic programming the atoms represent subsets of a fixed universe X and one is allowed to compose the one-step consequence operator with a monotonic idempotent operator O so as to ensure that the analogue of stable models in the theory are always closed under O. Marek et al. (1992, Ann Pure Appl Logic 96:231---276 1999) developed a generalization of Reiter's normal default theories that can be applied to both default theories and logic programs which is based on an underlying consistency property. In this paper, we show how to extend the normal logic programming paradigm of Marek, Nerode, and Remmel to set based logic programming. We also show how one can obtain a new semantics for set based logic programming based on a consistency property.

On the normal ideals of exchange rings

An ideal I of a ring R is called normal if all idempotent elements in I lie in the center of R. We prove that if I is a normal ideal of an exchange ring R then: (1) R and R/I have the same stable range; (2) V(I) is an order-ideal of the monoid C(Specc(R), N), where Specc(R) consists of all prime ideals P such that R/P is local.

On the finitistic dimension conjecture of Artin algebras

Let A be an Artin algebra and e be an idempotent element of A. We prove that if A has representation dimension at most three, then the finitistic dimension of eAe is finite, and deduce that if quasi-hereditary algebras have representation dimensions at most three, then the finitistic dimension conjecture holds.

SEMIGROUP PROPERTIES OF N-ARY OPERATIONS ON FINITE SETS

A clone is a set of operations defined on a base set A which is closed under composition and contains all the projection operations. There are several ways to regard a clone as an algebraic structure (see e.g. [3]). If f, g1,…,gn: An→ A are n-ary operations defined on A, then by Sn(f, g1… , gn)(a1… , an) := f(g1(a1,…,an),…,gn(a1,…,an)) for all a1,…, an∈ A an (n + 1)-ary operation on the set On(A) of all n-ary operations can be defined. From this operation one can derive a binary operation + defined by f + g := Sn(f, g,…,g) and obtains a semigroup (On(A);+). The collection of all clones of operations on a finite set forms a complete lattice. This lattice is well-described ([4], [5]) if |A| = 2. If |A| > 2, this lattice is uncountably infinite and very complex. In this paper instead of clones we study semigroups of n-ary operations, i.e. subsemigroups of the semigroup (On(A); +) and their properties. We look for idempotent and regular elements of (On(A); +), consider Green's relations for the semigroup (On(A); +), characterize all constant subsemigroups of (On(A);+), all semilattices, rectangular bands and normal bands contained in (On(A);+).

All idempotent hypersubstitutions of type (2,2)

A hypersubstitution of type (2,2) is a map σ which takes the binary operation symbols f and g to binary terms σ(f) and σ(g). Any such σ can be inductively extended to a map \(\hat{\sigma}\) on the set of all terms of type (2,2). By using this extension on the set Hyp(2,2) of all hypersubstitutions of type (2,2) a binary operation can be defined. Together with the identity hypersubstitution mapping f to f(x1,x2) and g to g(x1,x2) the set Hyp(2,2) forms a monoid. This monoid is isomorphic to the endomorphism monoid of the clone of all binary terms of type (2,2). We determine all idempotent elements of this monoid. The results can be applied to the equational theory of Universal Algebra.

On the finitistic dimension conjecture, III: Related to the pair [formula omitted

Let A be an Artin algebra and e an idempotent element in A. In this paper, we use co-homological conditions on A to control the finitistic dimension of eAe. Such a consideration is of particular interest for understanding the finitistic dimension conjecture. Let us denote the finitistic dimension and global dimension of A by fin.dim ( A ) and gl.dim ( A ) , respectively. Suppose gl.dim ( A ) ⩽ 4 . Then fin.dim ( e A e ) < ∞ if one of the following conditions holds: (1) A / A e A has representation dimension at most 3; (2) Ω A −3 ( A ) is an A / A e A -module; (3) proj.dim ( S A ) ⩽ 3 for all simple A / A e A -modules S. This result can be considered as a first step to the question of whether gl.dim ( A ) ⩽ 4 implies fin.dim ( e A e ) < ∞ . Moreover, we show the following: Let A be an arbitrary Artin algebra and e an idempotent element of A such that the ∗-syzygy dimension or the Gorenstein dimension of the eAe-module Ae is finite. If fin.dim ( A ) < ∞ , then fin.dim ( e A e ) < ∞ .

The Order of Generalized Hypersubstitutions of Typeτ=(2)

The order of hypersubstitutions, all idempotent elements on the monoid of all hypersubstitutions of typeτ=(2)were studied by K. Denecke and Sh. L. Wismath and all idempotent elements on the monoid of all hypersubstitutions of typeτ=(2,2)were studied by Th. Changpas and K. Denecke. We want to study similar problems for the monoid of all generalized hypersubstitutions of typeτ=(2). In this paper, we use similar methods to characterize idempotent generalized hypersubstitutions of typeτ=(2)and determine the order of each generalized hypersubstitution of this type. The main result is that the order is 1,2 or infinite.

Numerical Radius Inequalities for Square-zero and Idempotent Operators

We show that if A is a square-zero or an idempotent operator on a Hilbert space and B commutes with A ,t henw(AB) min{w(A)� B� , � Aw(B)} holds, where w(·) and �·� denote, respectively, the numerical radius and operator norm of an operator

Applications of polar decompositions of idempotent and 2-nilpotent operators†

An operator means a bounded linear operator on a complex Hilbert space. Using two polar decompositions of idempotent and 2-nilpotent operators, we shall study numerical radii of these two operators and finally we shall discuss two operator transformations, one of which is the generalized Aluthge transformation and another is an extension of the operator transformation by Patel–Tanahashi–Uchiyama. †This work is dedicated to Professor Masanori Fukamiya on his 95th birthday with respect and affection.

Local Structure of Algebraic Monoids

We describe the local structure of an irreducible algebraic monoid M at an idempotent element e. When e is minimal, we show that M is an induced variety over the kernel MeM (a homogeneous space) with bre the two-sided stabilizer Me (a connected a ne monoid having a zero element and a dense unit group). This yields the irreducibility of stabilizers and centralizers of idempotents when M is normal, and criteria for normality and smoothness of an arbitrary monoid M . Also, we show that M is an induced variety over an abelian variety, with ber a connected a ne monoid having a dense unit group.