Integral and differential of fractional order are important notion that is often used in dealing with Frechet geometry. Based on PSEUDO-operations given by monotone and continuous function g, we study PSEUDO-derivative and PSEUDO-integral of fractional order through this paper.

In this paper, the notion of generalization of PSEUDO p-closure, denoted by gcl, is introduced and its related properties are investigated. The gcl of subalgebras and PSEUDO-ideals is discussed. Also, a necessary and sufficient condition for an element to be minimal; and for PSEUDO BCI-algebra to be nilpotent are given. It is proved that the set of all nilpotent elements of a PSEUDO BCI-algebra A, denoted by NA, is the least closed PSEUDO-ideal with the property gcl(NA)=NA. Finally, it is shown that the mentioned notion, as a function, defines a closure operation on PSEUDO-ideals.

In this paper, persents the definitions of strongly prime ideal, strongly prime N-subgroup, PSEUDO-valuation near ring and PSEUDO-valuation N-group. Some of their properties have also been proven by theorems. Then it is shown that, if N be near ring with quotient near-field K and P be a strongly prime ideal of near ring N, then is a strongly prime ideal of , for any multiplication subset S of N. In addition, they obtained the relation between strongly prime ideal and strongly prime N-group, and also between PSEUDO-valuation near ring and PSEUDO-valuation N-subgroup. It has also shown that if every N-subgroup be ideal of M and P be a strongly prime N-subgroup of M, then (P: M) is a strongly prime ideal of N. And in the end it is proved that if P and L of N-subgroups M and Psubset of L such that for any y in K , y-1P subset of P, then L is a strongly prime N-subgroup of M if and only if L/p is a strongly prime N-subgroup of M/p.

The aim of this paper is to introduce the notion of commutative PSEUDO BE-algebras and investigate their properties. We generalize some results proved by A. Walendziak for the case of commutative BE-algebras. We prove that the class of commutative PSEUDO BE-algebras is equivalent to the class of commutative PSEUDO BCK-algebras. Based on this result, all results holding for commutative PSEUDO BCK-algebras also hold for commutative PSEUDO BE-algebras. For example, any finite commutative PSEUDO BE-algebra is a BE-algebra, and any commutative PSEUDO BE-algebra is a join-semilattice. Moreover, if a commutative PSEUDO BE-algebra is a meet-semilattice, then it is a distributive lattice. We define the pointed PSEUDO-BE algebras, and introduce and study the relative negations on pointed PSEUDO BE-algebras. Based on the relative negations we construct two closure operators on a PSEUDO BE-algebra. We also define relative involutive PSEUDO BE-algebras, we investigate their properties and prove equivalent conditions for a relative involutive PSEUDO BE-algebra. We define the relative Glivenko property for a relative good PSEUDO BE-algebra and show that any relative involutive PSEUDO BE-algebra has the relative Glivenko property.