Document Type : Research Paper


Department of Mathematics, University of Maragheh, Maragheh, 55181-83111, Iran.


Let $\mathcal M$ be a  class of (mono)morphisms in a category $\mathcal A$. To study mathematical notions, such as injectivity, tensor products, flatness, one needs to have some categorical and algebraic information about the pair (${\mathcal A}$,${\mathcal M}$).
In this paper, we take $\mathcal A$ to be the category {\bf Pos}-$S$ of $S$-posets over a posemigroup $S$, and ${\mathcal M}_{dc}$ to be the class of down closed embeddings and study the categorical properties, such as limits and colimits, of the  pair (${\mathcal A}$,${\mathcal M}$). Injectivity with respect to this class of monomorphisms have been studied by Shahbaz et al., who used it to obtain  information about regular injectivity.


[1] C.T. Shieh and V.A. Yurko, Inverse nodal and inverse spectral problems for discontinuous boundary value problems, J. Math. Anal. Appl., 347 (2008), pp. 266-272.
[2] G. Teschl, Mathematical Methods in Quantum Mechanics; With Applications to Schr"odinger Operators, Graduate Studies in Mathematics, Amer. Math. Soc., Rhode Island, 2009.
[3] J. Li, M. Yasuda and J. Song, Regularity properties of null-additive fuzzy measure on metric space, in: Proc. 2nd Internatinal Conference on Modeling Decisions for Artificial Intelligencer, Tsukuba, Japan, 2005, pp. 59-66.
[4] S. Bulman-Fleming, Subpullback flat $S$-posets need not be subequlizer flat, Semigroup Forum, 78 (1) (2009), pp. 27-33.
[5] S. Bulman-Fleming, D. Gutermuth, A. Gilmour and M. Kilp, Flatness properties of $S$-posets, Comm. Alg., 34 (4) (2006), pp. 1291-1317.
[6] S. Bulman-Fleming and V. Laan, Lazard's theorem for $S$-posets, Math. Nachr., 278 (15) (2005), pp. 1743-1755.
[7] S. Bulman-Fleming and M. Mahmoudi, The category of S-posets, Semigroup Forum, 71 (3) (2005), pp. 443-461.
[8] M. Mehdi Ebrahimi and M. Mahmoudi, The category of $M$-sets, Ital. J. Pure Appl. Math., 9 (2001), pp. 123-132.
[9] M. Mehdi Ebrahimi, M. Mahmoudi and H. Rasouli, Banaschewski's theorem for S-posets: regular injectivity and completeness, Semigroup Forum, 80 (2) (2010), pp. 313-324.
[10] S.M., Fakhruddin, On the category of S-posets, Acta Sci. Math. (Szeged), 52 (1988), pp. 85-92.
[11] M. Kilp, U. Knauer and A. Mikhalev, Monoids, Acts and Categories, Walter de Gruyter, Berlin, New York, 2000.
[12] H. Rasouli, Categorical properties of regular monomorphisms of $S$-posets, Eur. J. Pure Appl. Math., 7(2) (2014), pp. 166-178.
[13] L. Shahbaz and M. Mahmoudi, Various kinds of regular injectivity for $S$-posets, Bull. Iranian Math. Soc., 40 (1) (2014), pp. 243-261.
[14] L. Shahbaz and M. Mahmoudi, Injectivity of $S$-posets with respect to down closed regular monomorphisms, Semigroup Forum, 91 (2015), pp. 584-600.
[15] X. Shi, On flatness properties of cyclic $S$-posets, Semigroup Forum, 77 (2) (2008), pp. 248-266.
[16] L.A. Skornjakov, On the injectivity of ordered left acts over monoids, Vestnik Moskov. univ. Ser. I Math. Mekh., (1986), pp. 17-19 (in Russian).
[17] X. Zhang, Regular injectivity of S-posets over clifford pomonoids, Southeast Asian Bulletin of Math., 32 (2007), pp. 1007-1015.
[18] X. Zhang and V. Laan, On homological classification of pomonoids by regular weak injectivity properties of S-posets, Central Eur. J. Math., 5 (1) (2007), pp. 181-200.