Some classifications of monoids by strongly idempotent cancellative $\bf (PWP)$

Document Type : Research Paper

Authors

1 Department of Mathematics, University of Sistan and Baluchestan, Zahedan, Iran

2 Department of mathematics, University of Sistan and Baluchestan, Zahedan, Iran

Abstract

In this paper, we introduce Condition $(PWP_{sic})$ of acts over monoids and compare this condition with the properties left $PP$ and left $PSF$ in monoid $S$. At first we give a classification of monoids by this condition of right acts. Also, we give a classification of monoids for which some other properties of their right acts imply Condition $(PWP_{sic})$ and vice versa. Then a classification of monoids will be given for which all right Rees factor acts of $S$ satisfying some other flatness properties have Condition $(PWP_{sic})$.
The specific question of when every right $S$-act satisfying Condition $(PWP_{sic})$ has certain flatness properties or every ($GPW$-flat) $GP$-flat right $S$-act satisfies Condition $(PWP_{sic})$, have so far not been considered. In this paper, we will address these problems.

Keywords


[1] S. Bulman-Fleming, M. Kilp and V. Laan, Pullbacks and flatness properties of acts II, Comm. Algebra, 29 No. 2 (2001) 851-878.
[2] A. Golchin and H. Mohammadzadeh, On condition (EP), Int. Math. Forum, 2 No. 19 (2007) 911-918.
[3] A. Golchin and H. Mohammadzadeh, On condition (E′P), J. Sci. Islam. Repub. Iran, 17 No. 4 (2006) 343-349.
[4] A. Golchin and H. Mohammadzadeh, On condition (PWPE), Southeast Asian Bull. Math., 33 (2009) 245-256.
[5] A. Golchin and H. Mohammadzadeh, On homological classification of monoids by condition (PE) of right acts, Ital. J. Pure Appl. Math., 25 (2009) 175-186.
[6] A. Golchin and H. Mohammadzadeh, On condition (P′), Semigr. Forum, 86 (2013) 413-430.
[7] A. Golchin, A. Zare and H. Mohammadzadeh, E-torsion free acts over monoids, Thai J. Math., 19 No. 4 (2015) 93-114.
[8] J. M. Howie, Fundamentals of Semigroup Theory, London Mathematical Society Monographs, Oxford University Press, London, 1995.
[9] P. Khamechi, H. Mohammadzadeh Saany and L. Nouri, Classification of monoids by Condition (PWPssc) of right acts, Categ. Gen. Algebr. Struct. Appl., 12 No. 1 (2020) 175-197.
[10] M. Kilp and U. Knauer, On torsionless and dense acts, Semigr. Forum, 63 No. 3 (2001) 396-414.
[11] M. Kilp, U. Knauer and A. Mikhalev, Monoids, Acts and Categories, Walter de Gruyter, Berlin, 2000.
[12] V. Laan, Pullbacks and Flatness Properties of Acts, Ph.D. Thesis, Tartu, Estonia, 1999.
[13] V. Laan, Pullbacks and flatness properties of acts I, Comm. Algebra, 29 No. 2 (2001) 829-850.
[14] X. Liang and Y. Luo, On a generalization of weak pullback flatness, Comm. Algebra, 44 (2016) 3796-3817.
[15] H. Qiao and C. Wei, On a generalization of principal weak flatness property, Semigroup Forum, 85 (2012) 147-159.
[16] H. Rashidi, A. Golchin and H. Mohammadzadeh Saany, On GPW-flat acts, Categ. Gen. Algebr. Struct. Appl., 12 No. 1 (2020) 25-42.
[17] M. Sedaghatjoo, R. Khosravi and M. Ershad, Principally weakly and weakly coherent monoids, Comm. Algebra, 37 No. 12 (2009) 4281-4295.
[18] A. Zare, A. Golchin and H. Mohammadzadeh, R-torsion free acts over monoids, J. sci. Islam. Repub. Iran, 24 No. 3 (2013) 275-285.