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.

In this paper the chain length of a space of fuzzy orderings is defined, and various properties of this invariant are proved. The structure theorem for spaces of finite chain length is proved. Spaces of Fuzzy Orderings Throughout X = (X,A) denoted a space of fuzzy orderings. That is, A is a fuzzy subgroup of abelian group G of exponent 2. (see [1] (i.e. x 2 = 1,  x  G), and X is a (non empty) fuzzy subset of the character group  (A) = Hom(A,{1,–1}) satisfying: 1. X is a fuzzy closed subset of  (A). 2.  an element e  A such that (e) = – 1    X. 3. X :={a  A\ (a) = 1    X} = 1. 4. If f and g are forms over A and if x  D(

P. aeruginosa is one of the complex targets for antimicrobial chemotherapy. Also, it is intrinsically resistant to several antibiotics. It produces β-lactamases enzymes that are responsible for the widespread β-lactam antimicrobial resistance. There are three major groups of β-lactamase enzymes, MBLs and ESBLs forming Pseudomonas is a major issue for the treatment of burns victims. Methods: A total of 28 clinical isolates related to P. aeruginosa have been obtained from the burns specimens from patients attending to AL-Imam hospital/Baghdad-Iraq, through the period from October 2015 to March 2016. Also, all isolates have been recognized as P. aeruginosa via utilizing bacteriological assay and confirmed by Vitek 2. In addition, the suscep

In this ˑwork, we present theˑ notion of the ˑgraph for a KU-semigroup as theˑundirected simple graphˑ with the vertices are the elementsˑ of and weˑˑstudy the ˑgraph ofˑ equivalence classesˑofˑ which is determinedˑ by theˑ definition equivalenceˑ relation ofˑ these verticesˑ, andˑ then some related ˑproperties areˑ given. Several examples are presented and some theorems are proved. Byˑ usingˑ the definitionˑ ofˑ isomorphicˑ graph, ˑwe showˑ thatˑ the graphˑ of equivalence ˑclasses ˑand the ˑgraphˑof ˑa KU-semigroup ˑ areˑ theˑ sameˑ, in special cases.

      In this Ë‘work, we present theË‘ notion of the Ë‘graph for a KU-semigroup  as theË‘undirected simple graphË‘ with the vertices are the elementsË‘ of  and weË‘Ë‘study the Ë‘graph ofË‘ equivalence classesË‘ofË‘  which is determinedË‘ by theË‘ definition equivalenceË‘ relation ofË‘ these verticesË‘, andË‘ then some related Ë‘properties areË‘ given. Several examples are presented and some theorems are proved. ByË‘ usingË‘ the definitionË‘ ofË‘ isomorphicË‘ graph, Ë‘we showË‘ thatË‘ the graphË‘ of equivalence Ë‘classes Ë‘and the Ë‘graphË‘of Ë‘a KU-semigroup Ë‘  areË‘ theË‘ sameË‘,

     Given an exterior algebra  over a finite dimension vector space v, and let   , where  is a graded ideal in . The relation between the algebra  and    regarding to -quadratic and - quadratic will be investigated. We show that the algebra   is - quadratic if and only if   is - quadratic. Furthermore, it has been shown that the algebra  is - quadratic if and only if  is - quadratic.

The multiplicity of connotations in any paper does not mean that there is no main objective for that paper and certainly one of these papers is our research the main objective is to introduce a new connotation which is type-2 fuzzy somewhere dense set in general type-2 fuzzy topological space and its relationship with open sets of the connotation type-2 fuzzy set in the same space topology and theories of this connotation.

In this work, we present the notion of a multiplier on AT-algebra and investigate several properties. Also, some theorems and examples are discussed.  The notions of the kernel and the image of multipliers are defined. After that, some propositions related to isotone and regular multipliers are proved. Finally, the Left and the Right derivations of the multiplier are obtained