Idempotency Research Articles

On wreath product of n-polygroups

In this paper, we shall introduce thin n-subpolygroups of a given n-polygroup and in this regards, the notion of wreath product of n-polygroups will be studied. Also, double cosets of n-polygroups are investigated and the classical isomorphism theorems of groups are generalized to n-polygroups. The main result of the paper is that a finite n-polygroup is singular if and only if it is a wreath product of n-subpolygroups all of which are thin or generated by an involution or by an idempotent element.

Anisimov's Theorem for inverse semigroups

The idempotent problem of a finitely generated inverse semigroup is the formal language of all words over the generators representing idempotent elements. This paper proves that a finitely generated inverse semigroup with regular idempotent problem is necessarily finite. This answers a question of Gilbert and Noonan Heale, and establishes a generalization to inverse semigroups of Anisimov's Theorem for groups.

On the Cross-Migrativity with Respect to Continuous t-Norms

Depending on whether α is an idempotent element of the t-norm T0 or not, we study and characterize the structure of a continuous crossmigrative t-norm T with respect to a fixed and continuous t-norm T0, which is indeed the conjecture presented by Fodor eti¾?al. Int J Intell Syst 2012;27:411-428 Results of this paper show that the α,T0-cross-migrativity is completely determined by the restriction on a portion of domain of T and have nothing to do with the remaining of [0, 1]2. Moreover, we also clarify the relations between our present results and the previous results.

Idempotent and Regular Elements of the Complete Semigroups of Binary Relations of the Class ∑<sub>3</sub>(<i>X</i>,9)

In this paper, we take Q16 subsemilattice of D and we will calculate the number of right unit, idempotent and regular elements α of BX (Q16) satisfied that V (D, α) = Q16 for a finite set X. Also we will give a formula for calculate idempotent and regular elements of BX (Q) defined by an X-semilattice of unions D.

P*-closure operator and p*-regularity in fuzzy setting

In this paper a new type of fuzzy regularity, viz. fuzzy p*- regularity has been introduced and studied by a newly defined closure operator, viz., fuzzy p*-closure operator. Also we have found the mutual relationship of this closure operator among other closure operators defined earlier. In p*-regular space, p*-closure operator is an idempotent operator. In the last section, p*-closure operator has been characterized via p*-convergence of a fuzzy net.

Simple Logics for Basic Algebras

An MV-algebra is an algebra (A, ⊕, ¬, 0), where (A, ⊕, 0) is a commutative monoid and ¬ is an idempotent operation on A satisfying also some additional axioms. Basic algebras are similar algebras that can roughly be characterised as nonassociative (hence, also non-commutative) generalizations of MV-algebras. Basic algebras and commutative basic algebras provide an equivalent algebraic semantics in the sense of Blok and Pigozzi for two recent logical systems. Both are Hilbert-style systems, with implication and negation as the primitive connectives. We present a considerably simpler logic, Lʙ, for basic algebras, where implication and falsum are taken as primitives. We also consider some subvarieties of basic algebras known in the literature, discuss classes of implicational algebras term-equivalent to each of these varieties, and construct axiomatic extensions of Lʙ for which these classes serve as equivalent algebraic semantics.

Regular Elements of the Complete Semigroups <i>B<sub>X</sub>(D)</i> of Binary Relations of the Class ∑<sub>2</sub>(<i>X</i>,8)

As we know if D is a complete X-semilattice of unions then semigroup Bx(D) possesses a right unit iff D is an XI-semilattice of unions. The investigation of those a-idempotent and regular elements of semigroups Bx(D) requires an investigation of XI-subsemilattices of semilattice D for which V(D,a)=Q∈∑2(X,8) . Because the semilattice Q of the class ∑2(X,8) are not always XI -semilattices, there is a need of full description for those idempotent and regular elements when V(D,a)=Q . For the case where X is a finite set we derive formulas by calculating the numbers of such regular elements and right units for which V(D,a)=Q .

Complete Semigroups of Binary Relations Defined by Semilattices of the Class ∑<sub>1</sub>(<i>X</i>,10)

In this paper we give a full description of idempotent elements of the semigroup BX (D), which are defined by semilattices of the class ∑1 (X, 10). For the case where X is a finite set we derive formulas by means of which we can calculate the numbers of idempotent elements of the respective semigroup.

Enumeration of idempotents in diagram semigroups and algebras

We give a characterisation of the idempotents of the partition monoid, and use this to enumerate the idempotents in the finite partition, Brauer and partial Brauer monoids, giving several formulae and recursions for the number of idempotents in each monoid as well as various R-, L- and D-classes. We also apply our results to determine the number of idempotent basis elements in the finite dimensional partition, Brauer and partial Brauer algebras.

Dualizing clones into categories of topological spaces

In this paper, we study clones of dual operations in the category of topological spaces. Our investigation is motivated by the possibility to apply duality theory in order to obtain some information about clones in the classical sense. In this way, we deduce results about the centralizer clones of (not necessarily finite) Boolean algebras, C*-algebras and bounded distributive lattices. We examine their arity sequences, discuss the structure of the lattices given by their subclones, and take a closer look at their idempotent operations.

Algebraic and topological results on lifting properties in residuated lattices

We define lifting properties for universal algebras, which we study in this general context and then particularize to various such properties in certain classes of algebras. Next we focus on residuated lattices, in which we investigate lifting properties for Boolean and idempotent elements modulo arbitrary, as well as specific kinds of filters. We give topological characterizations to the lifting property for Boolean elements and several properties related to it, many of which we obtain by means of the reticulation.

S*-CLOSURE OPERATOR AND s*-REGULARITY IN FUZZY SETTING

In this paper a new type of fuzzy regularity, viz. fuzzy s � -regularity has been introduced and studied by introducing a newly defined closure opera- tor, viz.,s � -closure operator. Also we have found the mutual relationship among other closure operators defined earlier. In Section 4, it has been shown that s � -closure operator is an idempotent operator in fuzzy s � -regular space and in the last section some characterizations of s � -closure operator have been done

Maps Preserving Complementarity of Closed Subspaces of a Hilbert Space

AbstractLet and be infinite-dimensional separable Hilbert spaces and Lat the lattice of all closed subspaces oh . We describe the general form of pairs of bijective maps ϕ, Ψ : Lat → Lat having the property that for every pair U, V ∊ Lat we have . Then we reformulate this theorem as a description of bijective image equality and kernel equality preserving maps acting on bounded linear idempotent operators. Several known structural results for maps on idempotents are easy consequences.

The Order of Normal Form Generalized Hypersubstitutions of Type τ = (2)

In 2000, K. Denecke and K. Mahdavi showed that there are many idempotent elements in HypNφ(V ) the set of normal form hypersubstitutions of type = (2) which are not idempotent elements in Hyp(2) the set of all hypersubstitutions of type = (2). They considered in which varieties, idempotent elements of Hyp(2) are idempotent elements of HypNφ(V ). In this paper, we study the similar problems on the set of all generalized hypersubstitutions of type = (2) and the set of all normal form generalized hypersubstitutions of type = (2) and determine the order of normal form generalized hypersubstitutions of type = (2).

Retracted: Idempotent Elements of the Endomorphism Semiring of a Finite Chain

Retracted: Idempotent Elements of the Endomorphism Semiring of a Finite Chain

Construction of Rota-Baxter algebras via Hopf module algebras

We propose the notion of Hopf module algebras and show that the projection onto the subspace of coinvariants is an idempotent Rota-Baxter operator of weight -1. We also provide a construction of Hopf module algebras by using Yetter-Drinfeld module algebras. As an application, we prove that the positive part of a quantum group admits idempotent Rota-Baxter algebra structures.

An evolution algebra in population genetics

We consider an evolution algebra which corresponds to a bisexual population with a set of females partitioned into finitely many different types and the males having only one type. We study basic properties of the algebra. This algebra is commutative (and hence flexible), not associative and not necessarily power-associative, in general. We prove that being alternative is equivalent to being associative. We find conditions to be an associative, a fourth power-associative, or a nilpotent algebra. We also prove that if the algebra is not alternative then to be power-associative is equivalent to be Jordan. Moreover it is not unital. In a general case, we describe the full set of idempotent elements and the full set of absolute nilpotent elements. The set of all operators of left (right) multiplications is described. Under some conditions it is proved that the corresponding algebra is centroidal. Moreover the classification of 2-dimensional and some 3-dimensional algebras are obtained.

Strong commutativity preserving maps on rings

Suppose $\R$ is a unital ring having an idempotent element $e$ which satisfies $a{\R}e=0$ implies $a=0$ and $a{\R}(1-e)=0$ implies $a=0$. In this paper, we aim to characterize the map $f:{\R}\rightarrow {\R}$, $f$ is surjective and $[f(x),f(y)]=[x,y]$ for all $x,y\in {\R}$. It is shown that $f(x)=\alpha x +\xi (x)$ for all $x\in {\R}$, where $\alpha \in {\Z}({\R})$, $\alpha^2=1$, and $\xi$ is a map from ${\R}$ into ${\Z}({\R})$. As an application, a characterization of nonlinear surjective maps preserving strong commutativity on von Neumann algebras with no central summands of type $I_1$ is obtained.

An Idempotent Approach to Truncated Moment Problems

We present a new approach to truncated and full moment problems, via idempotent elements with respect to associated square positive Riesz functionals. The existence of representing measures for such functionals is characterized via some intrinsic conditions.

On real chains of evolution algebras

In this paper, we define a chain of -dimensional evolution algebras corresponding to a permutation of numbers. We show that a chain of evolution algebras (CEA) corresponding to a permutation is non-trivial if and only if the permutation has a fixed point. We show that a CEA is a chain of nilpotent algebras (independently on time) if it is trivial. We construct a wide class of chains of three-dimensional EAs and a class of symmetric -dimensional CEAs. A construction of arbitrary dimensional CEAs is given. Moreover, for a chain of three-dimensional EAs, we study the behaviour of the baric property, the behaviour of the set of absolute nilpotent elements and dynamics of the set of idempotent elements depending on the time.