# On a generalization of condition (PWP)

Document Type: Research Paper

Authors

1 Department of Mathematics‎, ‎Lanzhou University‎, ‎Lanzhou‎, ‎Gansu 730000‎, ‎P.R. China.‎ ‎

2 Department of Mathematics‎, ‎Lanzhou University‎, ‎Lanzhou‎, ‎Gansu 730000‎, ‎P.R. China.

Abstract

‎There is a flatness property of acts over monoids called Condition $(PWP)$ which‎, ‎so far‎, ‎has received‎ ‎much attention‎. ‎In this paper‎, ‎we introduce Condition GP-$(P)$‎, ‎which is a generalization of Condition $(PWP)$‎. ‎Firstly‎, ‎some  characterizations of monoids by Condition GP-$(P)$ of their‎ ‎(cyclic‎, ‎Rees factor) acts are given‎, ‎and many known results are generalized‎. ‎Moreover‎, ‎some possible conditions on monoids that describe when their diagonal acts satisfy Condition GP-$(P)$ are found‎. ‎Finally‎, ‎using some new types of epimorphisms‎, ‎an alternative description of Condition GP-$(P)$ (resp.‎, ‎Condition $(PWP)$) is obtained‎, ‎and directed‎ ‎colimits of these new epimorphisms are investigated.

Keywords

Main Subjects

### References

A. Bailey and J. Renshaw, Covers of acts over monoids and pure epimorphisms, Proc. Edinb. Math. Soc. (2) 57 (2014), no. 3, 589--617.

S. Bulman-Fleming, Flat and strongly at S-systems, Comm. Algebra 20 (1992), no. 9, 2553--2567.

S. Bulman-Fleming, M. Kilp and V. Laan, Pullbacks and atness properties of acts II, Comm. Algebra 29 (2001), no. 2, 851--878.

S. Bulman-Fleming and A. Gilmour, Flatness properties of diagonal acts over monoids, Semigroup Forum 79 (2009), no. 2, 298--314.

A. Golchin and H. Mohammadzadeh, On condition (P), Semigroup Forum 86 (2013), no. 2, 413--430.

J. M. Howie, Fundamentals of Semigroup Theory, London Mathematical Society Monographs, Oxford University Press, New York, 1995.

M. Kilp, On at acts, (Russian) Tartu Riikl.  Ul. Toimetised Vih. 253 (1970) 66--72.

M. Kilp, Commutative monoids all of whose principal ideals are projective, Semigroup Forum 6 (1973), no. 4, 334--339.

M. Kilp, Characterization of monoids by properties of their left Rees factors, (Russian) Tartu Riikl. Ul. Toimetised Vih. 640 (1983) 29--37 .

M. Kilp, U. Knauer and A. V. Mikhalev, Monoids, Acts and Categories, Walter de Gruyter & Co., Berlin, 2000.

V. Laan, Pullbacks and atness properties of acts I, Comm. Algebra 29 (2001), no. 2, 829--850.

Z. K. Liu and Y. B. Yang, Monoids over which every at right act satisfies condition (P), Comm. Algebra 22 (1994), no. 8, 2861--2875.

P. Normak, On equalizer-at and pullback-at acts, Semigroup Forum 36 (1987), no. 3, 293--313.

H. S. Qiao, Some new characterizations of right cancellative monoids by condition (PWP), Semigroup Forum 71 (2005), no. 1, 134--139.

H. S. Qiao, On a generalization of principal weak atness property, Semigroup Forum 85 (2012), no. 1, 147--159.

M. Sedaghatjoo, V. Laan and M. Ershad, Principal weak atness and regularity of diagonal acts, Comm. Algebra 40 (2012), no. 11, 4019--4030.

B. Stenstrom, Flatness and localization over monoids, Math. Nachr. 48 (1971) 315--334.

### History

• Receive Date: 16 January 2015
• Revise Date: 23 June 2015
• Accept Date: 23 June 2015