Independence algebras (Q1891269)

The author studies the endomorphism monoid of independence algebras. By an independence algebra she means an algebra where the subalgebra closure operator satisfies the Exchange Property and endomorphisms can be arbitrarily prescribed on any basis. Notable examples for independence algebras are: vector spaces, sets, free \(G\)-sets. In an independence algebra the rank of a subalgebra is defined as the cardinality of any basis of the subalgebra. In the endomorphism monoid of an independence algebra the endomorphisms with image of rank \(\leq n\) form an ideal \(T_n\). It is shown here that \(T_n /T_{n-1}\) is a completely 0-simple semigroup, and a Rees matrix representation for \(T_n/T_{n-1}\) is given both in general and in the three particular cases.