Strongly Flat andPO-FlatS-Posets
Xiaoping Shi · Communications in Algebra · 2005
For a monoid S, a (left) S-act is a nonempty set B together with a mapping S × B → B sending (s, b) to sb such that s(tb) = (st)b and 1b = b for all s, t ∈ S and b ∈ B. Over the past three decades, an extensive theory of flatness properties has been developed (involving free acts, projective acts, strongly flat acts, Condition (P), flat acts, weakly flat acts, principally weakly flat acts, and torsion free acts). A recent and complete discussion of this area is contained in the monograph Monoids, Acts and Categories by Kilp et al. (2000 Kilp , M. , Knauer , U. , Mikhalev , A. V. ( 2000 ). Monoids, Acts and Categories, with Applications to Wreath Products and Graphs . Berlin , New York : Walter de Gruyter .[Crossref] , [Google Scholar]). Partially ordered acts over a partially ordered monoid S, or S-posets appear naturally in the study of mappings between posets. Preliminary work on flatness properties of S-poset, was done by Fakhruddin in the 1980s (see Fakhruddin, 1986 Fakhruddin , S. M. ( 1986 ). Absolute flatness and amalgams in pomonoids . Semigroup Forum . 33 : 15 – 22 . [CSA] [Crossref], [Web of Science ®] , [Google Scholar] 1988 Fakhruddin , S. M. ( 1988 ). On the category of S-posets . Acta Sci. Math. (Szeged) . 52 : 85 – 92 . [CSA] [Google Scholar]), and continued in recent (Bulman-Fleming and Laan, 2005 Bulman-Fleming , S. , Laan , V. ( 2005 ). Lazard's Theorem for S-posets . Math. Nachr. 278 : 1 – 13 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]; Shi et al., 2005 Shi , X. P. , Liu , Z. K. , Wang , F. G. , Bulman-Fleming , S. ( 2005 ). Indecomposable, projective and flat S-posets . Comm. Algebra. 33 : 235 – 251 . [CSA] [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). In Bulman-Fleming and Laan (2005 Bulman-Fleming , S. , Laan , V. ( 2005 ). Lazard's Theorem for S-posets . Math. Nachr. 278 : 1 – 13 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]), the Stenström-Govorov-Lazard theorem was shown in the context of S-posets. Tensor products of S-posets, free, projective and flat S-posets, as well as an analogue of Condition (P) and Condition (E) were introduced in these articles, but strongly flat, flat, weakly flat, principally weakly flat acts and torsion free S-posets were not considered. The present article addresses these matters.