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.
BCIK-algebra, N-sub algebra (closed, commutative, retrenched) N-ideal, ?- negative functions, ?-translations, prime ideals, maximal ideals
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