Subpullbacks and coproducts of $S$-posets

Document Type: Research Paper

Authors

School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, PR China.

Abstract

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.

Keywords


[1] S. Bulman-Fleming, and V. Laan,  Lazard's theorem for $S$-posets, Math. Nachr. 278 (2005), 1-13.  
[2] S. Bulman-Fleming, M. Kilp, and V. Laan,  Pullbacks and flatness properties of acts II, Comm. Algebra 29(2) (2001), 851-878.  
[3] S. Bulman-Fleming, D. Gutermuth, A. Gilmour, and M. Kilp, Flatness properties of $S$-posets, Comm. Algebra 34 (2006), 1291-1317.  
[4] S. Bulman-Fleming, and M. Mahmoudi, The category of $S$-posets, Semigroup Forum 71 (2005), 443-461.  
[5] G. Cz$acute{e}$dli, and A. Lenkehegyi, On classes of ordered algebras and quasiorder distributivity, Acta Sci. Math. (Szeged) 46 (1983), 41-54.
[6] S. M. Fakhruddin, Absolute flatness and amalgams in pomonoids, Semigroup Forum 33 (1986), 15-22. 
[7]  S. M. Fakhruddin, On the category of $S$-posets, Acta Sci. Math. (Szeged) 52 (1998), 85-92.  
[8] A. Golchin, Flatness and coproducts, Semigroup Forum 72 (2006), 433-440.  
[9] A. Golchin, and P. Rezaei, Subpullbacks and flatness properties of $S$-posets, {Comm. Algebra} 37 (2009), 1995-2007.  
[10] V. Laan, Pullbacks and flatness properties of acts I, Comm. Algebra 29 (2001), 829-850.  
[11] X. P. Shi, On flatness properties of cyclic $S$-posets, {Semigroup Forum} 77 (2008), 248-266.  
[12] X. P. Shi, Z. K. Liu, F. G. Wang, and S. Bulman-Fleming, Indecomposable, projective, and flat $S$-posets, Comm. Algebra 33 (2005), 235-251.  
[13] X. Y. Xie, and X. P. Shi,  Order-congruences on $S$-posets, Comm. Korean Math. Soc. 20 (2005), 1-14.