Classification of monoids by Condition $(PWP_{ssc})$

Document Type : Research Paper

Authors

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

10.29252/cgasa.12.1.175

Abstract

Condition $(PWP)$ which was introduced in (Laan, V., {\it Pullbacks and flatness properties of acts I}, Commun. Algebra, 29(2) (2001), 829-850), is related to flatness concept of acts over monoids. Golchin and Mohammadzadeh in ({\it On Condition $(PWP_E)$}, Southeast Asian Bull. Math., 33 (2009), 245-256) introduced Condition $(PWP_E)$, such that Condition $(PWP)$ implies it, that is, Condition $(PWP_E)$ is a generalization of Condition $(PWP)$.

In this paper we introduce Condition $(PWP_{ssc})$, which is much easier to check  than Conditions $(PWP)$ and $(PWP_E)$ and does not imply them. Also principally weakly flat is a generalization of this condition. At first, general properties of Condition $(PWP_{ssc})$ will be given. Finally a classification of monoids will be given for which all (cyclic, monocyclic) acts satisfy Condition $(PWP_{ssc})$ and also a classification of monoids $S$ will be given for which all right $S$-acts satisfying some other flatness properties have Condition $(PWP_{ssc})$.

Keywords

References

[1] Bulman-Fleming, S., Flat and strongly flat S-systems, Comm. Algebra 20(9) (1992), 2553-2567.
[2] Bulman-Fleming, S. and Gilmour, A., Flatness properties of diagonal acts over monoids, Semigroup Forum, 79 (2009), 298-314.
[3] Bulman-Fleming, S., Kilp, M., and Laan, V., Pullbacks and flatness properties of acts II, Comm. Algebra 29(2) (2001), 851-878.
[4] Bulman-Fleming, S. and McDowell, K., A characterization of left cancellative monoids by flatness properties, Semigroup Forum 40 (1990), 109-112.
[5] Fountain, J., Right PP monoids with central idempotents, Semigroup Forum 13 (1977), 229-237.
[6] Golchin, A. and Mohammadzadeh, H., On Condition (P0), Semigroup Forum 86 (2013), 413-430.
[7] Golchin, A. and Mohammadzadeh, H., On homological classification of monoids by Condition (PE) of right acts, Ital. J. Pure Appl. Math. 25 (2009), 175-186.
[8] Golchin, A. and Mohammadzadeh, H., On Condition (PWPE), Southeast Asian Bull. Math. 33 (2009), 245-256.
[9] Golchin, A. and Mohammadzadeh, H., On Condition (EP), Int. Math. Forum 2(19) (2007), 911-918.
[10] Golchin, A. and Mohammadzadeh, H., On Condition (E0P), J. Sci. Islam. Repub. Iran 17(4) (2006), 343-349.
[11] Golchin, A. and Renshaw, J., A flatness property of acts over monoids, Conference on Semigroups, University of St. Andrews (1997-1998), 72-77.
[12] Howie, J.M., “Fundamentals of Semigroup Theory”, London Math. Soc. Monographs, Oxford University Press, 1995.
[13] Renshaw, J. and Golchin, A., Flat acts that satisfy Condition (P), Semigroup Forum 59 (1999), 295-309.
[14] Kilp, M., Knauer, U., and Mikhalev, A., “Monoids, Acts and Categories”, Walter de Gruyter, Berlin, 2000.
[15] Laan, V., “Pullbacks and Flatness Properties of Acts”, Ph.D. Thesis, Tartu, 1999.
[16] Laan, V., Pullbacks and flatness properties of acts I, Comm. Algebra 29(2) (2001), 829-850.
[17] Liu, Z.K. and Yang, Y.B., Monoids over which every flat right act satisfies Condition (P), Comm. Algebra 22(8) (1994), 2861-2875.
[18] Qiao, H.S., Some new characterizations of right cancellative monoids by Condition (PWP), Semigroup Forum 71 (2005), 134-139.
[19] Qiao, H.S., Wang, L.M., and Liu, Z.K., On some new characterizations of right cancellative monoids by flatness properties, Arab. J. Sci. Eng. 32 (2007), 75-82.
[20] Qiao, H.S. and Wei, C., On a generalization of principal weak flatness property, Semigroup Forum 85 (2012), 147-159.
[21] Zare, A., Golchin, A., and Mohammadzadeh, H., R-torsion free acts over monoids, J. Sci. Islam. Repub. Iran 24(3) (2013), 275-285.
[22] Zare, A., Golchin, A., and Mohammadzadeh, H., Strongly torsion free acts over monoids, Asian-Eur. J. Math. 6(3) (2013), 1-22.
[23] Sedaghatjoo, M., Khosravi, R., and Ershad, M., Principally weakly and weakly coherent monoids, Comm. Algebra 37 (2009), 4281-4295.
Categories and General Algebraic Structures with Applications
Volume 12, Issue 1
January 2020
Pages 175-197

Files

Share

How to cite

Statistics