Properties of products for flatness in the category of $S$-posets

Document Type: Research Paper


1 Department of Mathematics, Fasa University, Fasa, Iran.

2 Department of Mathematics, College of Science, Persian Gulf University, Bushehr, Iran.


This paper is devoted to the study of products of classes of right $S$-posets possessing one of the flatness properties and preservation of such properties under products. Specifically, we characterize a pomonoid $S$ over which its nonempty products as right $S$-posets satisfy some known flatness properties. Generalizing this results, we investigate products of right $S$-posets satisfying Condition $(PWP)$. Finally, we investigate pomonoids over which products of right $S$-posets transfer an arbitrary flatness property, projectivity, freeness, and regularity to their components.


[1] S. Bulman-fleming, D. Gutermuth, A. Glimour, and M. Kilp, Flatness properties of
S-posets, Comm. Algebra 34(4) (2006), 1291-1317.
[2] S. Bulman-Fleming and A. Gilmour, Flatness properties of diagonal acts over monoids,
Semigroup Forum 79(2) (2009), 298-314.
[3] S. Bulman-Fleming and K. McDowell, Coherent monoids, In: Almeida, J., et al., eds.
Latices, Semigroups and Universal Algebra. New York: Plenum Press, (1990).
[4] S. Bulman-Fleming, Products of projective S-system, Comm. Algebra 19(3) (1991),
[5] S. Bulman-Fleming and M. Mahmoudi, The category of S-posets, Semigroup Forum
71(3) (2005), 443-461.
[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 (1988),
[8] A. Golchin and P. Rezaei, Subpullbacks and flatness properties of S-posets, Comm.
Algebra 37(6) (2009), 1995-2007.
[9] R. Khosravi, On direct products of S-posets satisfying flatness properties, Turk. J.
Math. 38 (2014), 79-86.
[10] X.L. Liang and Y.F. Luo, On Condition (PWP)w for S-posets, Turk. J. Math. 39
(2015), 795-809.
[11] X.L. Liang and Y. Luo, Subpullbacks and coproducts of S-posets, Categ. General Alg.
Struct. Appl. 3(1) (2015), 1-20.
[12] L. Nouri, A. Golchin, and H. Mohammadzadeh, On Properties of Product acts over
monoids, Comm. Algebra 43(5) (2015), 1854-1876.
[13] H.S. Qiao and H.C. Wen, On a generalizations of principally weakly po-flat poset, J.
Shandong. Univ. 50(12) (2015), 1571-9352.

[14] M. Sedaghatjoo, R. Khosravi, and M. Ershad, Principaly weakly and weakly coherent
monoids, Comm. Algebra 37(12) (2009), 4281-4295.
[15] X. Shi, On flatness properties of cyclic S-posets, Semigroup Forum 77(2) (2008),
[16] X. Shi, Strongly flat and po-flat S-posets, Comm. Algebra 33(12) (2005), 4515-4531.
[17] X. Shi, Z. Liu, F. Wang, and S. Bulman-Fleming, Indecomposable, projective, and
flat S-posets, Comm. Algebra 33(1) (2005), 235-251.