On a generalization of condition (PWP)

Document Type : Research Paper


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

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


‎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.


Main Subjects

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.