Products of idempotent endomorphisms of an independence algebra of infinite rank

BY JOHN FOUNTAI ANND ANDREW LEWINDepartment of Mathematics, University of York, Heslington, York YOl 5DD(Received 12 August 1992; revised December 22 1992)AbstractIn 1966, J. M. Howie characterized the self-maps of a set which can be written as aproduct (under composition) of idempotent self-maps of the same set. In 1967, J. A.Erdos considered the analogous question for linear maps of a finite dimensionalvector space and in 1985, Reynolds and Sullivan solved the problem for linear mapsof an infinite dimensional vector space. Using the concept of independence algebra,the authors gave a common generalization of the results of Howie and Erdos for thecases of finite sets and finite dimensional vector spaces. In the present paper weintroduce strong independence algebras and provide a common generalization of theresults of Howie and Reynolds and Sullivan for the cases of infinite sets and infinitedimensional vector spaces.IntroductionFor a mathematical structure M the set of endomorphisms oiM, which we denoteby End(.M), is a monoid under composition of mappings. We let E denote the set ofnon-identity idempotents of End(M). Over the last twenty five years considerableeffort has been devoted to describing the subsemigroup (Ey generated by E. The firstresults were obtained by Howie in [5] where a set-theoretic description of (E} isgiven when M is simply a set and End(ilf) = T(M) is the full transformationsemigroup on M. To describe when if is an infinite set, we define, for a in T(M),C(oc) = {xeM:\(xcc)a-