Let Covering be the category of fuzzy coverings, and Partition, partitions. We geometrically construct an isomorphism categories between Partition a full subcategory Covering, which can used to derive bijections partitions coverings with finitely many sets. Also, we establish Covering[n], n sets, whose objects are sets satisfy certain conditions.

in this paper we define an order structure on the $p$-operator projective tensor product of herz algebras and we show that the canonical isometric isomorphism between $a_p(gtimes h)$ and $a_p(g)widehat{otimes}^p a_p(h)$ is an order isomorphism for amenable groups $g$ and $h$.

Let R = ⊕ g∈GRg be a G-graded ring. It is well known (see e.g. [D], [M1], [N], [NRV], [NV]) that in the study of the connections that may be established between the categories R-gr of graded R-modules and R1-mod (1 is the unit element of G), an important role is played by the following system of functors : (−)1 : R-gr → R1-mod given by M 7→ M1, where M = ⊕ g∈GMg is a graded left R-module, the i...

Let Mmn =Mmn(F) denote the set of all m× n matrices over a field F, and fix some n×m matrix A ∈ Mnm. An associative operation ? may be defined on Mmn by X ? Y = XAY for all X,Y ∈ Mmn, and the resulting sandwich semigroup is denoted Mmn = Mmn(F). In this article, we study Mmn as well as its subsemigroups Reg(Mmn) and EA mn (consisting of all regular elements and products of idempotents, respecti...

There is a 2-category J -Colim of small categories equipped with a choice of colimit for each diagram whose domain J lies in a given small class J of small categories, functors strictly preserving such colimits, and natural transformations. The evident forgetful 2-functor from J -Colim to the 2-category Cat of small categories is known to be monadic. We extend this result by considering not jus...

Proof: For f ∈ Hom(X,A), i ◦ f = 0 implies (i ◦ f)(x) = 0 for all x ∈ X, and then f(x) = 0 for all x since i is injective. Thus, Hom(X,A)→ Hom(X,B) is injective, giving exactness at the left joint. Since q ◦ i = 0, any f ∈ Hom(X,A) is mapped to 0 ∈ Hom(X,C) by f → q ◦ i ◦ f . That is, the image of i ◦ − is contained in the kernel of q ◦ −. On the other hand, when g ∈ Hom(X,B) is mapped to q ◦ g...

Interest in weak cubical n-categories arises in various contexts, in particular in topological field theories. In this paper, we describe a concept of double bicategory in terms of bicategories internal to Bicat. We show that in a special case one can reduce this to what we call a Verity double bicategory, after Domenic Verity. This is a weakened version of a double category, in the sense that ...

X id → X → 0→ · For any morphism u : X → Y , there is an object Z (called a mapping cone of the morphism u) fitting into a distinguished triangle X u − → Y → Z → · Any triangle isomorphic to a distinguished triangle is distinguished. This means that if X u − → Y v − → Z w −→ X[1] is a distinguished triangle, and f : X → X, g : Y → Y , and h : Z → Z are isomorphisms, then X′ gu f −1 −−−−→ Y ′ hv...

Abstract We prove that if the Auslander–Reiten triangles generate relations for Grothendieck group of a Hom-finite Krull–Schmidt triangulated category with (co)generator, then has only finitely many isomorphism classes indecomposable objects up to translation. This gives converse theorem Butler and on groups. Our approach applications in context Frobenius categories.