Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-58533120150701Subpullbacks and coproducts of $S$-posets1208992ENXingliang LiangSchool of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu
730000, PR China.Yanfeng LuoSchool of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, PR China.Journal Article20150406In 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 acts $A_{S}$ over a monoid $S$ that can be described by means of when the functor $A_{S} otimes -$ preserves pullbacks. In this paper, we extend these results to $S$-posets and present equivalent descriptions of weakly kernel po-flat, principally weakly kernel po-flat and translation kernel po-flat. Moreover, we show that most of flatness properties of $S$-posets can be transferred to their coproducts and vice versa.Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-58533120150701Actions of a separately strict cpo-monoid on pointed directed complete posets214210031ENHalimeh Moghbeli DamanehShahid Beheshti UniversityJournal Article20150704 In the present article, we study some categorical properties of the category {$bf Cpo_{Sep}$-$S$} of all {separately strict $S$-cpo's}; cpo's equipped with a compatible right action of a separately strict cpo-monoid $S$ 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 in {$bf Cpo_{Sep}$-$S$} are studied, and show that {$bf Cpo_{Sep}$-$S$} is not cartesian closed. Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-58533120150701Order dense injectivity of $S$-posets436310518ENLeila ShahbazDepartment of Mathematics, University of MaraghehJournal Article20150822In this paper, the notion of injectivity with respect to order dense embeddings in the category of $S$-posets, posets with a monotone action of a pomonoid $S$ on them, is studied. We give a criterion, like the Baer condition for injectivity of modules, or Skornjakov criterion for injectivity of $S$-sets, for the order dense injectivity. Also, we consider such injectivity for $S$ 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 of$S$-posets.Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-58533120150701$omega$-Operads of coendomorphisms and fractal $omega$-operads for higher structures658810527ENCamell KachourDepartment of Mathematics, Macquarie University, Sydney, Australia.Journal Article20150824 In this article we introduce the notion of textit{Fractal $omega$-operad} emerging from a natural $omega$-operad associated to any coglobular object in the category of higher operads in Batanin's sense, which in fact is a coendomorphism $omega$-operads. We have in mind coglobular object of higher operads which algebras are kind of higher transformations. It follows that this natural $omega$-operad acts on the globular object associated to these higher transformations. To construct the natural $omega$-operad we introduce some general technology and give meaning to saying an $omega$-operad possesses the textit{fractal property}. If an $omega$-operad $B^{0}_{P}$ has this property then one can define a globular object of all higher $B^{0}_{P}$-transformations and show that the globular object has a $B^{0}_{P}$-algebra structure.Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-58533120150701Operads of higher transformations for globular sets and for higher magmas8911110528ENCamell KachourDepartment of Mathematics, Macquarie University, Sydney, Australia.Journal Article20150824In this article we discuss examples of fractal $omega$-operads. Thus we show that there is an $omega$-operadic approach to explain existence of the globular set of globular setsfootnote{Globular sets are also called $omega$-graphs by the French School.}, the reflexive globular set of reflexive globular sets, the $omega$-magma of $omega$-magmas, and also the reflexive $omega$-magma of reflexive $omega$-magmas. Thus, even though the existence of the globular set of globular sets is intuitively evident, many other higher structures which textit{fractality} are less evident, could be described with the same technology, using fractal $omega$-operads. We have in mind the non-trivial question of the existence of the weak $omega$-category of the weak $omega$-categories in the globular setting, which is described in cite{kach-ir3} with the same technology up to a contractibility hypothesis.Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-58533120150701A cottage industry of lax extensions11315110709ENDirk HofmannDepartamento de Matem ́atica, Universidade de Aveiro, 3810-193 Aveiro, Portugal.Gavin J. SealEcole Polytechnique F ́ed ́erale de Lausanne, Station 8, CH-1015 Lausanne, SwitzerlandJournal Article20150907In 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.