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 semigroups have (split) basic semigroup algebras, two-sided semidirect product decompositions of finite monoids; unambiguous products of rational languages; products of rational languages with counter; and Černý's conjecture for an important class of automata.

We survey a great number of results obtained during four decades of investigations on lattices of semigroup varieties and formulate several open problems.

A nonlocal chiral quark model is consistently extended beyond mean field using a strict 1/Nc expansion scheme. The parameters of the nonlocal model are refitted so that the physical values of the pion mass and the weak pion decay constant are obtained. The size of the 1/Nc correction to the quark condensate is carefully studied and compared with the usual local Nambu\char21{}Jona-Lasinio model. It is found that even the sign of the corrections can be different. This can be attributed to the mesonic cutoff of the local model. The model is also applied to finite temperature. We find that the 1/Nc corrections dominate the melting of the chiral condensate at low temperatures, T$\lesssim${}100\,\,MeV, in agreement with chiral perturbation theory. On the other hand, the relative importance of the 1/Nc corrections in the crossover regime depends on the parameter T0 of the Polyakov-loop potential. For T0=270\,\,MeV, corresponding to a fit of lattice data for pure gluodynamics, the correction terms are large and lead to a lowering of the chiral phase-transition temperature in comparison with the mean-field result. Near the phase transition the 1/Nc expansion breaks down and a nonperturbative scheme is needed to include mesonic correlations in that regime. Lowering T0 leads to a more rapid crossover even at the mean-field level and the unstable region for the 1/Nc corrections shrinks. For T0$\lesssim${}220\,\,MeV the temperatures of deconfinement and chiral restoration are practically synchronized.