We prove that the non-commutative perspective of an operator convex function is the unique extension of the corresponding commutative perspective that preserves homogeneity and convexity

By studying and using the quasi-pure part concept, we im-prove some statements and show that some assumptions in some articles are super uous. We give some characterizations of Gelfand rings. For example: we prove that R is Gelfand if and only if m (Σ,, 2A I,) Σ,= , 2A m(I, ), for each family fI, g, 2A of ideals of R, in addition if R is semiprimitive and Max(R) ,Y ,Spec(R), we show that R is a Gelfand ring if and only if Y is normal. We prove that if R is reduced ring, then R is a von Neumann regular ring if and only if Spec(R) is regular. It has been shown that if R is a Gelfand ring, then Max(R) is a quotient of Spec(R), and sometimes hM(a)'s behave like the zerosets of the space of maximal ideal. Finally, it has been proven that Z ( Max(C(X)) ) = fhM(f): f 2 C(X)g if and only if fhM(f): f 2 C(X)g is closed under countable intersection if and only if X is pseudocompact.

In this paper, we introduce the notion of a block commu-tativity in several groupoids, and show that the class of block commutative groupoids and the class of d/BCK-algebras are Smarandache disjoint. The block commuta-tivity in linear/quadratic groupoids is investigated, and we prove that every group is a normal groupoid. More-over, we discuss block n-commutative groupoids and block ranks.

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.

In this paper, it is proven that a semigroup is regular and RGC_n-commutative if, and only if, it is a spined product of a commutative Clifford semigroup and a right regular band

It is shown that a commutative reduced ring R is a Baer ring if and only if it is a CS-ring; if and only if every dense subset of Spec (R) containing Max (R) is an extremally disconnected space; if and only if every non-zero ideal of R is essential in a principal ideal generated by an idempotent.

We consider pseudoquotient extensions of positive linear functionals on a commutative Banach algebra A and give conditions under which the constructed space of pseudoquotients can be identified with all Radon measures on the structure space Â.

The notion of a neutrosophic quadruple BCI-commutative ideal in a neutro-sophic quadruple BCI-algebra is introduced, and several properties are investigated. Rela-tions between a neutrosophic quadruple ideal and a neutrosophic quadruple BCI-commutative ideal are discussed, and conditions for the neutrosophic quadruple ideal to be a neutrosophic quadruple BCI-commutative ideal are given. Conditions for the neutrosophic quadruple set to be a neutrosophic quadruple BCI-commutative ideal are provided, and the extension property of a neutrosophic quadruple BCI-commutative ideal is considered.