the paper sets forth in detail categorically-algebraic or catalg foundations for the operations of taking the image and preimage of (fuzzy) sets called forward and backward powerset operators. motivated by an open question of s. e. rodabaugh, we construct a monad on the category of sets, the algebras of which generate the fixed-basis forward powerset operator of l. a. zadeh. on the next step, w...

BLOCKINy the construction of the PL-category in 3 directly from a given PL-category A without any reference to an ambient locally cartesian closed category L. An object in the bre over U n is a pair hT; T 0 i such that (i) T is an object in A(U n)|let t: U n-U be such that A(t)(X) = T |

This paper is to characterize L-concavities and L-convexities via some derived forms of relations operators. Specifically, notions L-concave internal relation space hull are introduced. It proved that the category L-concave
 spaces isomorphic spaces. Also, L-convex enclosed

Employing the notions of the strong $L$-topology introduced by Zhangand the $L$-frame introduced by Yao  and the concept of $L$-enrichedtopological system defined in the present paper, we constructadjunctions among the categories {bf St$L$-Top} of strong$L$-topological spaces, {bf S$L$-Loc} of strict $L$-locales and{bf $L$-EnTopSys} of $L$-enriched topological systems. All of theseconcepts are ...

Let $R$ be a commutative Noetherian ring, $fa$ anideal of $R$ and $mathcal{D}(R)$ denote the derived category of$R$-modules. For any homologically bounded complex $X$, we conjecture that$sup {bf L}Lambda^{fa}(X)leq$ mag$_RX$. We prove thisin several cases. This generalize the main result of Hatamkhani and Divaani-Aazar cite{HD} for complexes.

This paper presents a batch classifier that splits dataset into tree branches depending on the category type. It has been improved from earlier version and fixed mistake in paper. Two important changes have made. The first is to represent each with separate classifier. Each then classifies its own subset of data rows, using input values create centroid also itself. If contains more than one how...

Characterising colimiting ω-cocones of projection pairs in terms of least upper bounds of their embeddings and projections is important to the solution of recursive domain equations. We present a universal characterisation of this local property as ω-cocontinuity of locally continuous functors. We present a straightforward proof using the enriched Yoneda embedding. The proof can be generalised ...

Let G be a split connected reductive group over a finite field Fp with algebraic closure k, fix an Iwahori subgroup B ⊂ G(Fp([[t]])) and let F l = G(k((t)))/Bk denote the affine flag variety of G. Let q denote a power of p and fix a prime l 6= p. In [GH], the authors study the Jordan-Hölder series for objects in the Hecke category P q (F l,Ql). This is the category of B-equivariant perverse Wei...