N Sequence Prime Ideals

In this paper, the concepts of -sequence prime ideal and -sequence quasi prime ideal are introduced. Some properties of such ideals are investigated. The relations between -sequence prime ideal and each of primary ideal, -prime ideal, quasi prime ideal, strongly irreducible ideal, and closed ideal, are studied. Also, the ideals of a principal ideal domain are classified into quasi prime ideals and -sequence quasi prime ideals.

Derivations on the module extension Banach algebras

UDC 517.986 We correct some results presented in [M. Eshaghi Gordji, F. Habibian, A. Rejali, <em> Ideal amenability of module extension Banach algebras</em>, Int. J. Contemp. Math. Sci., <strong>2</strong>, No. 5, 213–219 (2007)] and, using the obtained consequences, we find necessary and sufficient conditions for the module extension to be -weakly amenable, where is a closed ideal of the Banach algebra and is a closed -submodule of the Banach -bimodule We apply this result to the module extension where are two Banach -bimodules.

When a locally compact monothetic semigroup is compact

A semigroup endowed with a topology is monothetic if it contains a dense monogenic subsemigroup. A semigroup (group) endowed with a topology is semitopological (quasitopological) if the translations (the translations and the inversion) are continuous. If S is a nondiscrete monothetic semitopological semigroup, then the set {S^{\prime}} of all limit points of S is a closed ideal of S. Let S be a locally compact nondiscrete monothetic semitopological semigroup. We show that (1) if the translations of {S^{\prime}} are open, then {S^{\prime}} is compact, and (2) if {S^{\prime}} can be topologically and algebraically embedded in a quasitopological group, then {S^{\prime}} is a compact topological group.

Event and Its Application in Algebraic Structures

The notion of eventful element and eventful set is introduced, and its application is considered in BCI and/or BCK algebras. Several properties are investigated, and (positive implicative) ideal in an eventful BCK algebra and closed ideal in an eventful BCI algebra are discussed. Given the subsets [Formula: see text] and [Formula: see text] of a BCI and/or BCK algebra, the eventful subset [Formula: see text] which consists of BCI and/or BCK events with a condition is established. Conditions for the eventful subset [Formula: see text] to be a (positive implicative) ideal of an eventful BCK algebra are provided, and conditions for the eventful subset [Formula: see text] to be a (closed) ideal of an event BCI algebra are given. An example to show that the set [Formula: see text] is not a positive implicative ideal in an eventful BCK algebra is provided, and then the conditions for the set [Formula: see text] to be a positive implicative ideal in an eventful BCK algebra are considered.

Primeness of Relative Annihilators in BCK-Algebra

Conditions that are necessary for the relative annihilator in lower B C K -semilattices to be a prime ideal are discussed. Given the minimal prime decomposition of an ideal A, a condition for any prime ideal to be one of the minimal prime factors of A is provided. Homomorphic image and pre-image of the minimal prime decomposition of an ideal are considered. Using a semi-prime closure operation “ c l ”, we show that every minimal prime factor of a c l -closed ideal A is also c l -closed.

Ideals of the Quantum Group Algebra, Arens Regularity and Weakly Compact Multipliers

Let $\mathbb{G}$ be a locally compact quantum group and let $I$ be a closed ideal of $L^{1}(\mathbb{G})$ with $y|_{I}\neq 0$ for some $y\in \text{sp}(L^{1}(\mathbb{G}))$. In this paper, we give a characterization for compactness of $\mathbb{G}$ in terms of the existence of a weakly compact left or right multiplier $T$ on $I$ with $T(f)(y|_{I})\neq 0$ for some $f\in I$. Using this, we prove that $I$ is an ideal in its second dual if and only if $\mathbb{G}$ is compact. We also study Arens regularity of $I$ whenever it has a bounded left approximate identity. Finally, we obtain some characterizations for amenability of $\mathbb{G}$ in terms of the existence of some $I$-module homomorphisms on $I^{\ast \ast }$ and on $I^{\ast }$.

ec-Filters and ec-ideals in the functionally countable subalgebra of C*(X)

<p>The purpose of this article is to study and investigate e_c-filters on X and e_c-ideals in C^*_c(X) in which they are in fact the counterparts of z_c-filters on X and z_c-ideals in C_c(X) respectively. We show that the maximal ideals of C^*_c(X) are in one-to-one correspondence with the e_c-ultrafilters on X. In addition, the sets of e_c-ultrafilters and z_c-ultrafilters are in one-to-one correspondence. It is also shown that the sets of maximal ideals of C_c(X) and C^*_c(X) have the same cardinality. As another application of the new concepts, we characterized maximal ideals of C^*_c(X). Finally, we show that whether the space X is compact, a proper ideal I of C_c(X) is an e_c-ideal if and only if it is a closed ideal in C_c(X) if and only if it is an intersection of maximal ideals of C_c(X).</p>

On (α,β)-US Sets in BCK/BCI-Algebras

Molodtsov originated soft set theory, which followed a general mathematical framework for handling uncertainties, in which we encounter the data by affixing the parameterized factor during the information analysis. The aim of this paper is to establish a bridge to connect a soft set and the union operations on sets, then applying it to B C K / B C I -algebras. Firstly, we introduce the notion of the ( α , β ) -Union-Soft ( ( α , β ) -US) set, with some supporting examples. Then, we discuss the soft B C K / B C I -algebras, which are called ( α , β ) -US algebras, ( α , β ) -US ideals, ( α , β ) -US closed ideals, and ( α , β ) -US commutative ideals. In particular, some related properties and relationships of the above algebraic structures are investigated. We also provide the condition of an ( α , β ) -US ideal to be an ( α , β ) -US closed ideal. Some conditions for a Union-Soft (US) ideal to be a US commutative ideal are given by means of ( α , β ) -unions. Moreover, several characterization theorems of (closed) US ideals and US commutative ideals are given in terms of ( α , β ) -unions. Finally, the extension property for an ( α , β ) -US commutative ideal is established.

Engel, Nilpotent and Solvable BCI-algebras

In this paper, we define the concepts of Engel, nilpotent and solvable BCI-algebras and investigate some of their properties. Specially, we prove that any BCK-algebra is a 2-Engel. Then we define the center of a BCI-algebra and prove that in a nilpotent BCI-algebra X, each minimal closed ideal of X is contained in the center of X. In addition, with some conditions, we show that every finite BCI-algebra is solvable. Finally, we investigate the relations among Engel, nilpotent and solvable BCI(BCK)-algebras.