Skip to main content Accessibility help
×
Home

Nilpotents and congruences on semigroups of transformations with fixed rank

  • M. Paula O. Marques-Smith (a1) and R. P. Sullivan (a2)

Abstract

In 1988, Howie and Marques-Smith studied Pm, a Rees quotient semigroup of transformations associated with a regular cardinal m, and described the elements which can be written as a product of nilpotents in Pm. In 1981, Marques proved that if Δm denotes the Malcev congruence on Pm, then Pmm is congruence-free for any infinite m. In this paper, we describe the products of nilpotents in Pm when m is nonregular, and determine all the congruences on Pm when m is an arbitrary infinite cardinal. We also investigate when a nilpotent is a product of idempotents.

Copyright

References

Hide All
1Clifford, A. H. and Preston, G. B.. The Algebraic Theory of Semigroups (Providence, RI: American Mathematical Society, Mathematical Surveys, No. 7, vols 1 and 2, 1961 and 1967).
2Howie, J. M.. The subsemigroup generated by the idempotents of a full transformation semigroup. J. London Math. Soc. 41 (1966), 707–16.
3Howie, J. M.. An Introduction to Semigroup Theory (London: Academic Press, 1976).
4Howie, J. M.. Some subsemigroups of infinite full transformation semigroups. Proc. Roy. Soc. Edinburgh Sect. A 88 (1981), 159–67.
5Howie, J. M. and Paula, M.Marques-Smith, O.. A nilpotent-generated semigroup associated with a semigroup of full transformations. Proc. Roy. Soc. Edinburgh Sect. A 108 (1988), 181–7.
6Thomas, Jech. Set Theory (New York: Academic Press, 1978).
7Malcev, A. I.. Symmetric groupoids. Mat Sb. 31 (1952), 136–51 (in Russian); English translation by Heidi Semla and Bob Sullivan: Amer. Math. Soc. Trans. 113 (1979), 235–50.
8Paula, M.Marques, O.. A congruence-free semigroup associated with an infinite cardinal number. Proc. Roy. Soc. Edinburgh Sect. A 93 (1983), 245–57.
9Monk, J. D.. Introduction to Set Theory (New York: McGraw-Hill, 1969).
10Sullivan, R. P.. Semigroups generated by nilpotent transformations. J. Algebra 110 (1987), 324–43.

Nilpotents and congruences on semigroups of transformations with fixed rank

  • M. Paula O. Marques-Smith (a1) and R. P. Sullivan (a2)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed