Shahid Beheshti University
Categories and General Algebraic Structures with Applications
2345-5853
2345-5861
3
1
2015
07
01
Subpullbacks and coproducts of $S$-posets
1
20
EN
Xingliang
Liang
School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu
730000, PR China.
lxl_119@126.com
Yanfeng
Luo
School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, PR China.
luoyf@lzu.edu.cn
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 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.
$S$-poset,subpullback,flatness,coproduct
https://cgasa.sbu.ac.ir/article_8992.html
https://cgasa.sbu.ac.ir/article_8992_c4f4ba1f7b36eaeed79d6c77c92e6f45.pdf
Shahid Beheshti University
Categories and General Algebraic Structures with Applications
2345-5853
2345-5861
3
1
2015
07
01
Actions of a separately strict cpo-monoid on pointed directed complete posets
21
42
EN
Halimeh
Moghbeli Damaneh
Shahid Beheshti University
h\_moghbeli@sbu.ac.ir
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.
Directed complete partially ordered set,Product,coproduct,cartesian closed
https://cgasa.sbu.ac.ir/article_10031.html
https://cgasa.sbu.ac.ir/article_10031_d6b54d9be2f2a6c5a9c0ff061f40bc88.pdf
Shahid Beheshti University
Categories and General Algebraic Structures with Applications
2345-5853
2345-5861
3
1
2015
07
01
Order dense injectivity of $S$-posets
43
63
EN
Leila
Shahbaz
Department of Mathematics, University of Maragheh
leilashahbaz@yahoo.com
In 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.
regular monomorphism,order dense sub $S$-poset,order dense injective
https://cgasa.sbu.ac.ir/article_10518.html
https://cgasa.sbu.ac.ir/article_10518_c47373537bc4095c9b7b11ef095abf14.pdf
Shahid Beheshti University
Categories and General Algebraic Structures with Applications
2345-5853
2345-5861
3
1
2015
07
01
$omega$-Operads of coendomorphisms and fractal $omega$-operads for higher structures
65
88
EN
Camell
Kachour
Department of Mathematics, Macquarie University, Sydney, Australia.
camell.kachour@gmail.com
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.
Higher categories,higher operads,weak higher transformations
https://cgasa.sbu.ac.ir/article_10527.html
https://cgasa.sbu.ac.ir/article_10527_391dbbec71f3cd9c77ef8d4d484a2ed9.pdf
Shahid Beheshti University
Categories and General Algebraic Structures with Applications
2345-5853
2345-5861
3
1
2015
07
01
Operads of higher transformations for globular sets and for higher magmas
89
111
EN
Camell
Kachour
Department of Mathematics, Macquarie University, Sydney, Australia.
camell.kachour@gmail.com
In 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.
Higher categories,higher operads,weak higher transformations
https://cgasa.sbu.ac.ir/article_10528.html
https://cgasa.sbu.ac.ir/article_10528_b04f19db4ee999d77afc297225d3cf14.pdf
Shahid Beheshti University
Categories and General Algebraic Structures with Applications
2345-5853
2345-5861
3
1
2015
07
01
A cottage industry of lax extensions
113
151
EN
Dirk
Hofmann
Departamento de Matem ́atica, Universidade de Aveiro, 3810-193 Aveiro, Portugal.
dirk@ua.pt
Gavin J.
Seal
Ecole Polytechnique F ́ed ́erale de Lausanne, Station 8, CH-1015 Lausanne, Switzerland
gavin.seal@fastmail.fm
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.
Monad,lax extension,quantale,enriched category
https://cgasa.sbu.ac.ir/article_10709.html
https://cgasa.sbu.ac.ir/article_10709_6dda525bdadbecfac60152d5865b84f9.pdf