CATEGORIES AND GENERAL ALGEBRAIC STRUCTURES WITH APPLICATIONS
Year:2015 | Volume:3 | Issue:1|Papers Count:6

In 2001, S. Bulman-Fleming et al. initiated the study of three flatness properties (weakly kernel flat, principally weakly kernel flat, translation kernel flat) of right actsAS over a monoid S that can be described by means of when the functorAS Ä- preserves pullbacks. In this paper, we extend these results toS-posets and present equivalent descriptions of weakly kernel po-flat, principally weakly kernel po-flat and translation kernel po-flat S-posets. Moreover, we show that most of flatness properties of S-posets can be transferred to their coproducts and vice versa.

In the present article, we study some categorical properties of the categoryCpoSep-S of all separately strict S-cpo’s; cpo’s equipped with a compatible right action of a separately strict cpo-monoidS which is strict continuous in each component. In particular, we show that this category is reflective and coreflective in the category of S-cpo’s, find the free and cofree functors, characterize products and coproducts. Furthermore, epimorphisms and monomorphisms inCpoSep-S are studied, and show that CpoSep-S is not cartesian closed.

In this paper, the notion of injectivity with respect to order dense embeddings in the category ofS-posets, posets with a monotone action of a pomonoidS on them, is studied. We give a criterion, like the Baer condition for injectivity of modules, or Skornjakov criterion for injectivity ofS-sets, for the order dense injectivity. Also, we consider such injectivity forS itself, and its order dense ideals. Further, we define and study some kinds of weak injectivity with respect to order dense embeddings, consider their relations with order dense injectivity. Also investigate if these kinds of injectivity are preserved or reflected by products, coproducts, and direct sums ofS-posets.

In this article we introduce the notion of Fractal w-operad emerging from a natural  w-operad associated to any coglobular object in the category of higher operads in Batanin’s sense, which in fact is a coendomorphism. w-operads. We have in mind coglobular object of higher operads which algebras are kind of higher transformations. It follows that this natural w-operad acts on the globular object associated to these higher transformations. To construct the natural w-operad we introduce some general technology and give meaning to saying an w-operad possesses the fractal property. If an w-operad B0P has this property then one can define a globular object of all higherB0P-transformations and show that the globular object has aB0P-algebra structure.

In this article we discuss examples of fractal w-operads. Thus we show that there is an w-operadic approach to explain existence of the globular set of globular sets1, the reflexive globular set of reflexive globular sets, the w-magma of w-magmas, and also the reflexive w-magma of reflexive.w-magmas. Thus, even though the existence of the globular set of globular sets is intuitively evident, many other higher structures which fractality are less evident, could be described with the same technology, using fractal w-operads. We have in mind the non-trivial question of the existence of the weak w-category of the weak w-categories in the globular setting, which is described in [9] with the same technology up to a contractibility hypothesis.

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.