In this paper we characterize the congruence associated to the direct sum of all irreducible representations of a finite semigroup over an arbitrary field, generalizing results of Rhodes for the field of complex numbers. Applications are given to obtain many new results, as well as easier proofs of several results in the literature, involving: triangularizability of finite semigroups; which sem...

We establish finiteand infinite-dimensional versions of the following assertion. If M is a matrix with the property that whenever P and Q are diagonal projections with P ≤ Q, the spectrum of PMP (considered as an operator on the range of P ) is contained in that of QMQ (considered as an operator on the range of Q), then there is a permutation matrix U such that U−1MU is triangular.