Normal Semigroups of Endomorphisms of Proper Independence Algebras are Idempotent Generated

Let X be a finite set. We denote by T (X) the monoid of all (total) transformations on X and by Sym(X) the symmetric group on X. An element a 2 T (X) \ Sym(X) is said to be singular. In 1966 Howie [4] proved that every singular transformation a 2 T (X) can be expressed as a product of idempotents of T (X). This result was generalized by Fountain and Lewin [2] for the case of independence algebras as follows. Let A be an independence algebra of finite §Universidade Aberta, R. da Escola Politécnica, 147, 1269-001 Lisboa, Portugal ([email protected]). †The author acknowledges with thanks the support of FCT, Programa Ciência, Tecnologia e Inovação do Quadro Comunitário de Apoio, and Fundação Calouste Gulbenkian.