[KSCI] Korea Science Citation Index Service

http://dx.doi.org/10.4134/JKMS.2015.52.1.067

REGULAR INJECTIVITY AND EXPONENTIABILITY IN THE SLICE CATEGORIES OF ACTIONS OF POMONOIDS ON POSETS

Farsad, Farideh (Department of Mathematics Semnan University)
Madanshekaf, Ali (Department of Mathematics Semnan University)

Publication Information

Journal of the Korean Mathematical Society / v.52, no.1, 2015 , pp. 67-80 More about this Journal

Abstract

For a pomonoid S, let us denote Pos-S the category of S-posets and S-poset maps. In this paper, we consider the slice category Pos-S/B for an S-poset B, and study some categorical ingredients. We first show that there is no non-trivial injective object in Pos-S/B. Then we investigate injective objects with respect to the class of regular monomorphisms in this category and show that Pos-S/B has enough regular injective objects. We also prove that regular injective objects are retracts of exponentiable objects in this category. One of the main aims of the paper is to draw attention to characterizing injectivity in the category Pos-S/B under a particular case where B has trivial action. Among other things, we also prove that the necessary condition for a map (an object) here to be regular injective is being convex and present an example to show that the converse is not true, in general.

Keywords

S-poset; slice category; regular monomorphism; injectivity;

Citations & Related Records

Reference

1	J. Adamek, H. Herrlich, J. Rosicky, and W. Tholen, Weak factorization systems and topological functors, Appl. Categ. Structures 10 (2002), no. 1, 237-249. DOI ScienceOn
2	A. Bailey and J. Renshaw, Weak factorization system for S-acts, Semigroup Forum 89 (2014), no. 1, 52-67. DOI ScienceOn
3	B. Banaschewski, Injectivity and essential extensions in equational classes of algebras, Proc. Conf. on Universal Algebra (Queen's Univ., Kingston, Ont., 1969) pp. 131-147 Queen's Univ., Kingston, Ont. 1970.
4	B. Banaschewski and G. Bruns, Categorical characterization of the MacNeille completion, Arch. Math. 18 (1967), 369-377. DOI
5	M. M. Ebrahimi, M. Mahmoudi, and H. Rasouli, Banaschewski's theorem for S-posets: regular injectivity and completeness, Semigroup Forum 80 (2010), no. 2, 313-324. DOI
6	G. Birkhoff, Lattice Theory, American Mathematical Society, Providence, 1973.
7	S. Bulman-Fleming and M. Mahmoudi, The category of S-posets, Semigroup Forum 71 (2005), no. 3, 443-461. DOI
8	F. Cagliari and S. Mantovani, Injectivity and sections, J. Pure Appl. Algebra 204 (2006), no. 1, 79-89. DOI ScienceOn
9	S. M. Fakhruddin, On the category of S-posets, Acta Sci. Math. (Szeged) 52 (1988), no. 1-2, 85-92.
10	M. Kilp, U. Knauer, and A. Mikhalev, Monoids, Acts and Categories, de Gruyter, Berlin, 2000.
11	S. Mac Lane and I. Moerdijk, Sheaves in Geometry and Logic: A First Introduction to Topos Theory, Springer Universitext, New York, 1992.
12	A. Madanshekaf and J. Tavakoli, Tiny objects in the category of M-Sets, Ital. J. Pure Appl. Math. 10 (2001), 153-162.
13	C. McLarty, Elementary Categories, Elementary Toposes, Oxford University Press, New York, 1992.
14	S. Niefield, Cartesianness: topological spaces, uniform spaces and affine schemes, J. Pure Appl. Algebra 23 (1982), no. 2, 147-167. DOI ScienceOn
15	S. Niefield, Exponentiable morphisms: posets, spaces, locals, and Grothendieck toposes, Theory Appl. Categ. 8 (2001), 16-32.
16	W. Tholen, Injectives, exponentials, and model categories, In: Abstracts of the Int. Conf. on Category Theory, 183-190, Como, Italy, 2000.