Semigroup Forum

, Volume 81, Issue 1, pp 102–127 | Cite as

Perfection for pomonoids

  • Victoria Gould
  • Lubna Shaheen
Research Article


A pomonoid S is a monoid equipped with a partial order that is compatible with the binary operation. In the same way that M-acts over a monoid M correspond to the representation of M by transformations of sets, S-posets correspond to the representation of a pomonoid S by order preserving transformations of posets.

Following standard terminology from the theories of R-modules over a unital ring R, and M-acts over a monoid M, we say that a pomonoid S is left poperfect if every left S-poset has a projective cover.

Left perfect rings were introduced in 1960 in a seminal paper of Bass (Trans. Am. Math. Soc. 95, 446–488, 1960) and shown to be precisely those rings satisfying M R , the descending chain condition on principal right ideals. In 1971, inspired by the results of Bass and Chase (Trans. Am. Math. Soc. 97, 457–473, 1960), Isbell was the first to study left perfect monoids Isbell (Semigroup Forum 2, 95–118, 1971). The results of Isbell, together with those of Fountain (Proc. Edinb. Math. Soc. 20, 87–93, 1976), show that a monoid is left perfect if and only if it satisfies a finitary condition dubbed Condition (A), in addition to M R . Moreover, M R can be replaced by another finitary condition, namely Condition (D).

A further characterisation of left perfect rings was given in Chase (1960), where Chase proved that a ring is left perfect if and only if every flat left module is projective; the corresponding result for M-acts was demonstrated in Fountain (1976).

In this paper we continue the study of left poperfect pomonoids, recently initiated in Pervukhin and Stepanova (Algebra Logic 48, 54–71, 2009). We show, as in Pervukhin and Stepanova, that a pomonoid S is left poperfect if and only if it satisfies (M R ) and the ‘ordered’ version Condition (Ao) of Condition (A) and further, these conditions are equivalent to every strongly flat left S-poset being projective. On the other hand, we argue via an analysis of direct limits that Conditions (A) and (Ao) are equivalent, so that a pomonoid S is left perfect if and only if it is left poperfect. We also give a characterisation of left poperfect monoids involving the ordered version of Condition (D). Our results and many of our techniques certainly correspond to those for monoids, but we must take careful account of the partial ordering on S, and in places introduce alternative strategies to those found in Isbell, Fountain and Pervukhin and Stepanova.


Perfect pomonoids Projective covers \(\mathcal{P}=\mathcal{SF}\) 


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  1. 1.
    Bass, H.: Finitistic dimension and a homological generalization of semi-primary rings. Trans. Am. Math. Soc. 95, 446–488 (1960) MathSciNetGoogle Scholar
  2. 2.
    Blyth, T.S., Janowitz, M.F.: Residuation Theory. Pergamon, Oxford (1972) zbMATHGoogle Scholar
  3. 3.
    Bulman-Fleming, S.: Flatness properties of S-posets: an overview. In: Proceedings of the International Conference on Semigroups, Acts and Categories, with Applications to Graphs, pp. 28–40. Estonian Mathematical Society, Tartu (2008) Google Scholar
  4. 4.
    Bulman-Fleming, S., Laan, V.: Lazard’s theorem for S-posets. Math. Nachr. 278, 1743–1755 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bulman-Fleming, S., Mahmoudi, M.: The category of S-posets. Semigroup Forum 71, 443–461 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Chase S.: Direct products of modules. Trans. Am. Math. Soc. 97, 457–473 (1960) MathSciNetGoogle Scholar
  7. 7.
    Clifford, A.H., Preston, G.: The Algebraic Theory of Semigroups, Vol. II. Math. Surveys, vol. 7. Amer. Math. Soc., Providence (1967) Google Scholar
  8. 8.
    Czedli, G., Lenkehegyi, A.: On classes of ordered algebras and quasiorder distributivity. Acta Sci. Math. (Szeged) 46, 41–54 (1983) zbMATHMathSciNetGoogle Scholar
  9. 9.
    Fahkruddin, S.M.: Absolute flatness and amalgams in pomonoids. Semigroup Forum 33, 15–22 (1986) CrossRefMathSciNetGoogle Scholar
  10. 10.
    Fahkruddin, S.M.: On the category of S-posets. Acta Sci. Math. (Szeged) 52, 85–92 (1988) MathSciNetGoogle Scholar
  11. 11.
    Fountain, J.B.: Perfect semigroups. Proc. Edinb. Math. Soc. 20, 87–93 (1976) zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Higgins, P.M.: Techniques of Semigroup Theory. Oxford Science Publications, Oxford (1992) zbMATHGoogle Scholar
  13. 13.
    Isbell, J.R.: Perfect monoids. Semigroup Forum 2, 95–118 (1971) zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Kilp, M.: On monoids over which all strongly flat cyclic right acts are projective. Semigroup Forum 52, 241–245 (1996) zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Kilp, M.: Perfect monoids revisited. Semigroup Forum 53, 225–229 (1996) zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Kilp, M., Knauer, U., Mikhalev, A.V.: Monoids, Acts, and Categories. de Gruyter, Berlin (2000) zbMATHGoogle Scholar
  17. 17.
    Laan, V., Zhang, X.: On homological classification of pomonoids by regular weak injectivity properties of S-posets. Cent. Eur. J. Math. 5, 181–200 (2007) zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Pervukhin, M.A., Stepanova, A.A.: Axiomatisability and completeness of some classes of partially ordered polygons. Algebra Logic 48, 54–71 (2009) CrossRefMathSciNetGoogle Scholar
  19. 19.
    Renshaw, J.: Monoids for which condition (P) acts are projective. Semigroup Forum 61, 46–56 (2000) zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Shaheen, L.: Axiomatisability problems for S-posets. Thesis (in preparation) Google Scholar
  21. 21.
    Shi, X.: Strongly flat and po-flat S-posets. Commun. Algebra 33, 4515–4531 (2005) zbMATHCrossRefGoogle Scholar
  22. 22.
    Shi, X.: On flatness properties of cyclic S-posets. Semigroup Forum 77, 248–266 (2007) CrossRefGoogle Scholar
  23. 23.
    Shi, X., Xie, X.Y.: Order-congruence on S-posets. Commun. Korean Math. Soc. 20 1–14 (2005) CrossRefMathSciNetGoogle Scholar
  24. 24.
    Shi, X., Liu, Z., Wang, F., Bulman-Fleming, S.: Indecomposable, projective, and flat S-posets. Commun. Algebra 33, 235–251 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Tajnia, S.: Projective covers in POS-S. Tarbiat Moallem University, 20th Seminar on Algebra, 2–3 Ordibhest 1388, 210–212 (Apr. 22–23, 2009) Google Scholar

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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of YorkYorkUK

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