In this paper, introduced a new notions of N sub algebras. closed, commutative, retrenched N ideals, negative functions, and translations are introduced, and related properties are investigated. Characterizations of an N sub algebra and a Commutative N ideal are given. Relations between an N sub algebra an N ideal and commutative N ideal are discussed. We verify that every translations of an N sub algebra resp. N ideal is a retrenched N sub algebra resp.retrenched N ideals . Already introduced a notationof prime ideal in BCIK – algebra show that it is equivalent to the last definition of prime ideal in lower BCIK – Semi lattices. Then we attempt to generalized some useful theorems about prime ideals, in BCIK algebras, instead of lower BCIK Semi lattices. We already investigate some results for PI lattices being a new classical of BCIK–lattices. Specially, we prove that any Boolean lattice is a PI lattice and we show that if X is a PI lattice with bounded commutative, then X is an involutory BCIK algebra if and only if X is a commutative BCIK algebra.
In this paper discuss and investigate a class of algebras which is related to several classes of algebras of interest such as BCIK algebras and which seems to have rather nice properties without being excessively complicated otherwise. A study on BK algebra discuss and investigate a class of algebras and define BK algebra and its properties and also see the Commutative Derived BK algebra. The notion of D algebras which is the another use full generalization of BCIK algebra. In this paper generalization of BCIK BK D algebras to introduce BD algebra and its properties.
In this paper we study the regularity of inside or outside derivations in p semi simple BCIK – algebra X and prove that let d X X be an inside derivation of X. If there exists a X such that d x a = 0, then d is regular for all x X. It is also show that if X is a BCIK algebra, then every inside or outside derivation of X is regular. Furthermore the concepts of ideal, ideal and invariant inside or outside derivation of X are introduced and their related properties are investigated. Finally we obtain the following result If d X X is an outside derivation of X, then d is regular if and only if every ideal of X is d invariant.
In the present paper, we introduced the notion of left derivation of a BCIK – algebra and investigate some related properties. Using the idea of regular left derivation of a BCIK algebra and investigate p semi simple BCIK algebra related properties. Using regular left derivation, we give characterizations of a regular left derivation on p semi simple BCIK algebra.In the present paper, we introduced the notion of left derivation of a BCIK – algebra and investigate some related properties. Using the idea of regular left derivation of a BCIK algebra and investigate p semi simple BCIK algebra related properties. Using regular left derivation, we give characterizations of a regular left derivation on p semi simple BCIK algebra.
