We construct a q-model structure, an h-model structure and an m-model structure on multipointed d-spaces and on flows. The two q-model structures are combinatorial and left determined and they coincide with the combinatorial model structures already known on these categories. The four other model structures (the two m-model structures and the two h-model structures) are accessible. We give an example of multipointed d-space and of flow which are not cofibrant in any of the model structures. We explain why the m-model structures, Quillen equivalent to the q-model structure of the same category, are better behaved than the q-model structures.

In this work, we describe an adjunction between the comma category of Set-based monads under the V -powerset monad and the category of associative lax extensions of Set-based monads to the category of V -relations. In the process, we give a general construction of the Kleisli extension of a monad to the category of V -relations.

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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We consider a category C enriched over the segment [0; 1] whose hom-objects are real numbers from [0; 1]. For a suitably defined function ^v assigning to each formula ' some object of C, the hom-object C(^v('); ^v( )) represents the degree of derivability ofψ from φ. We reformulate completeness result for intuitionistic propositional logic, as well as H ajek's completeness results concerning the product, G odel and Lukasiewicz fuzzy logic in the context of enriched category theory.

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.

For many centuries the linguists have almost always studied morphology. But with the advent of Transformational Grammar , syntactic studies and the structure of sentences have earned maximum importance, therefore Sentence was the focus of study until 70s , and from then on , little by little linguists have begun to study the structure of NP, and they have found many similarities between NPs , VPs and sentences . The idea of this similarity has led to many comparative studies. The Empty Category is one of the arguments discussed in sentence structure and pro /PRO is one of these empty categories. This article intends to indicate the existence of pro/PRO in NPs . Comparing Persian and Italian languages we find the same control properties in sentences.