In this paper a new type of fuzzy regularity, viz. fuzzy p∗regularity has been introduced and studied by a newly defined closure operator, viz., fuzzy p∗-closure operator. Also we have found the mutual relationship of this closure operator among other closure operators defined earlier. In p∗-regular space, p∗-closure operator is an idempotent operator. In the last section, p∗-closure operator h...

In this paper, we have focused to study convex $L$-subgroups of an $L$-ordered group. First, we introduce the concept of a convex $L$-subgroup and a convex $L$-lattice subgroup of an $L$-ordered group and give some examples. Then we find some properties and use them to construct convex $L$-subgroup generated by a subset $S$ of an $L$-ordered group $G$ . Also, we generalize a well known result a...

The aim of this paper is to introduce $(L,M)$-fuzzy closurestructure where $L$ and $M$ are strictly two-sided, commutativequantales. Firstly, we define $(L,M)$-fuzzy closure spaces and getsome relations between $(L,M)$-double fuzzy topological spaces and$(L,M)$-fuzzy closure spaces. Then, we introduce initial$(L,M)$-fuzzy closure structures and we prove that the category$(L,M)$-{bf FC} of $(L,M...

All GL (n) covariant star-body-valued valuations on convex polytopes are completely classified. It is shown that there is a unique nontrivial such valuation. This valuation turns out to be the so-called “intersection operator”—an operator that played a critical role in the solution of the Busemann-Petty problem. Introduction. A function Z defined on the set K of convex bodies (that is, of conve...

It is clear that if A7 possesses the spectral resolution N = jzdK(z), then any operator of the form A =Jzll2dK(z), where, for the value of z1'2, the choice of the branch of the function may depend on z, is a solution of (1). Moreover, all such operators are even normal. Of course, equation (1) may have other, nonnormal, solutions A. The object of this note is to point out a simple condition to ...

The following classical example of convex geometries shows how they earned their name. Given a set of points X in Euclidean space R, one defines a closure operator on X as follows: for any Y ⊆ X, Y = convex hull(Y ) ∩ X. One easily verifies that such an operator satisfies the anti-exchange axiom. Thus, (X,−) is a convex geometry. Denote by Co(R, X) the closure lattice of this closure space, nam...

Relativizing an idea from multiplicity theory, we say that an element x of a von Neumann algebra M is n-divisible if W (x) ∩ M unitally contains a factor of type In. We decide the density of the n-divisible operators, for various n, M, and operator topologies. The most sensitive case is σ-strong density in II1 factors, which is closely related to the McDuff property. We make use of Voiculescu’s...

The notion of pseudo-annulets is introduced in Stone lattices and characterized in terms of prime filters. Two operator α and β are introduced and obtained that their composition β ◦α is a closure operator on the class of all filters of a Stone lattice. A congruence θ is introduced on a Stone lattice L and proved that the quotient lattice L/θ is a Boolean algebra.