INDECOMPOSABLE, PROJECTIVE, AND FLATS-POSETS

Xiaoping Shi Department of Mathematics, Nanjing Forestry University, Nanjing, P. R. ChinaCorrespondence[email protected]

Zhongkui Liu Department of Mathematics, Northwest Normal University, Lanzhou, P. R. China

Fanggui Wang Department of Mathematics, Sichuan Normal University, Chengdu, P. R. China

Sydney Bulman-Fleming Department of Mathematics, Wilfrid Laurier University, Waterloo, Canada

Abstract

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) = lpar;st)b and 1b = b for all S , t ∈ S and B ∈ B. Right S -acts A can also be defined, and a tensor product A ⊗ _s B (a set)can be defined that has the customary universal property with respect to balanced maps from A × B into arbitrary sets. Over the past three decades, an extensive theory of flatness properties has been developed (involving free and projective acts, and flat acts of various sorts, defined in terms of when the tensor product functor has certain preservation properties). A recent and complete discussion of this area is contained in the monograph Monoids, Acts and Categories by M. Kilp et al. (New York: Walter de Gruyter, 2000). To date, there have been only a few attempts to generalize this material to ordered monoids acting on partially ordered sets ( S -posets). The present paper is devoted to such a generalization. A unique decomposition theorem for S -posets is given, based on strongly convex, indecomposable S -subposets, and a structure theorem for projective S -posets is given. A criterion for when two elements of the tensor product of S -posets given, which is then applied to investigate several flatness properties.

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ACKNOWLEDGMENTS

The first author was supported by the Specialized Research Fund for the Doctoral Program of Higher Education (20030284033), the second author was supported by the National Natural Science Foundation of China (10171082) and by the Teaching and Research Award Program for Outstanding Young Teachers in Higher Education Institutions of Ministry of Education, P.R. China, the third author was supported by the National Natural Science Foundation of China (1027105), and the fourth author supported by Natural Sciences and Engineering Council of Canada Discovery Grant A4494.

Notes

^#Communicated by Higgins.

^‡Dedicated to Professor Wenting Tong on the occasion of his 65th birthday.

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