J. Fountain et A. Lewin, PRODUCTS OF IDEMPOTENT ENDOMORPHISMS OF AN INDEPENDENCE ALGEBRA OF INFINITE RANK, Mathematical proceedings of the Cambridge Philosophical Society, 114, 1993, pp. 303-319
In 1966, J.M. Howie characterized the self-maps of a set which can be
written as a product (under composition) of idempotent self-maps of th
e same set. In 1967, J. A. Erdos considered the analogous question for
linear maps of a finite dimensional vector space and in 1985, Reynold
s and Sullivan solved the problem for linear maps of an infinite dimen
sional vector space. Using the concept of independence algebra, the au
thors gave a common generalization of the results of Howie and Erdos f
or the cases of finite sets and finite dimensional vector spaces. In t
he present paper we introduce strong independence algebras and provide
a common generalization of the results of Howie and Reynolds and Sull
ivan for the cases of infinite sets and infinite dimensional vector sp
aces.