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Journal of the London Mathematical Society Advance Access published online on June 10, 2009
Journal of the London Mathematical Society, doi:10.1112/jlms/jdp025
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© 2009 London Mathematical Society

The Bergman property for semigroups

V. Maltcev, J. D. Mitchell and N. Ruskuc

Mathematical Institute
University of St Andrews
North Haugh
St Andrews
Fife
KY16 9SS
United Kingdom
vm55@st-and.ac.uk
nr1@st-and.ac.uk

In this article, we study the Bergman property for semigroups and the associated notions of cofinality and strong cofinality. A large part of the paper is devoted to determining when the Bergman property, and the values of the cofinality and strong cofinality, can be passed from semigroups to subsemigroups and vice versa. Numerous examples, including many important semigroups from the literature, are given throughout the paper. For example, it is shown that the semigroup of all mappings on an infinite set has the Bergman property but that its finitary power semigroup does not; the symmetric inverse semigroup on an infinite set and its finitary power semigroup have the Bergman property; the Baer–Levi semigroup does not have the Bergman property.

2000 Mathematics Subject Classification 20M10 (primary), 20M20 (secondary).

Received December 21, 2007; revised February 6, 2009;
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