In this paper, we show that the CATEGORY of directed complete posets with bottom elements (cpos) endowed with an action of a monoid M on them forms a MONOIDAL CATEGORY. It is also proved that this CATEGORY is symmetric closed.

We give the concept of ‘braiding’ for 2-groupoids, and we show that this structure is equivalent to braided regular, crossed modules.

The aim of this paper is to introduce an abstract notion of determinant which we call quantum determinant, verifying the properties of the classical one. We introduce R−, basis and R−, solution on rigid objects of a MONOIDAL 𝐴, 𝑏, −, CATEGORY, for a compatibility relation R, such that we require the notion of duality introduced by Joyal and Street, the notion given by Yetter and Freyd and the classical one, then we show that R−, solutions over a semisimple ribbon 𝐴, 𝑏, −, CATEGORY form as well a semisimple ribbon 𝐴, 𝑏, −, CATEGORY. This allows us to define a concept of so-called quantum determinant in ribbon CATEGORY. Moreover, we establish relations between these and the classical determinants. Some properties of the quantum determinants are exhibited.

In this paper, it is shown that the CATEGORY of stratified L-generalized convergence spaces is MONOIDAL closed if the underlying truth-value table L is a complete residuated lattice. In particular, if the underlying truth-value table L is a complete Heyting Algebra, the Cartesian closedness of the CATEGORY is recaptured by our result.

We study cocompleteness, co-wellpoweredness, and generators in the centralizer CATEGORY of an object or morphism in a MONOIDAL CATEGORY, and the center or the weak center of a MONOIDAL CATEGORY. We explicitly give some answers for when colimits, cocompleteness, co-wellpoweredness, and generators in these MONOIDAL categories can be inherited from their base monidal categories. Most importantly, we investigate cofree objects of comonoids in these MONOIDAL categories.

The main goal of this paper is to study universal and existential quantifiers on involutive MONOIDAL t-norm based algebras, which are algebraic semantics for the logic of involutive left-continuous t-norms and their residua, and the resulting class of algebras will be called monadic IMTL-algebras. First we study some of their related algebraic properties and prove that the variety of monadic IMTL-algebras is the equivalent algebraic semantics of monadic predicate fuzzy logic $\mathbf{mMTL_{\forall}}$, which is equivalent to the modal fuzzy logic $\mathbf{S5(IMTL)}$, and show the completeness for $\mathbf{IMTL_{\forall}}$ via functional monadic IMTL-algebras. Moreover we start a systematic study of monadic algebraic structures that related to the monadic IMTL-algebras, some of which constitute the monadic MTL-algebras, monadic WNM-algebras, monadic NM-algebras, monadic BL-algebras, monadic MV-algebras and monadic Boolean algebras.Finally we give some representations of monadic IMTL-algebras. In particular, we character representable and directly indecomposable monadic IMTL-algebras by monadic filters.

We investigate and generalize quantum determinants to semisimple spherical and pivotal categories. It is well known that traces are preserved by strong tensor functors; we show on one hand that in fact, weaker conditions on a functor are sufficient to continue preserving traces. On the other hand, we prove that these determinants are well-behaved under strong tensor functors. Further, we introduce a notion of domination rank for objects of a semisimple pivotal CATEGORY and prove similar properties of the ordinary case. Furthermore, we expand the determinantal and McCoy ranks to introduce a morphism quantum rank function on a semisimple pivotal CATEGORY.

In this paper first we define the morphism between geometric spaces in two different types. We construct two categories of U-GESP and L-GESP from geometric spaces then investigate some properties of the two categories, for instance U-GESP is topological. The relation between hypergroups and geometric spaces is studied. By constructing the CATEGORY SNS-Hv of Hv-groups we answer the question that which construction of hyperstructures on the CATEGORY of sets has free object in the sense of universal property. At the end we define the CATEGORY of geometric Hypergroups and we study its relation with the CATEGORY of hypergroup.

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