A theory of transformation monoids: combinatorics and representation theory (Q612950)

This papers develops the theory of finite transformation monoids in the spirit similar to that of finite permutation groups. The main emphasis is made on the study of primitive transformation monoids. The paper consists of three parts: the first part systemizes the foundations of the theory, the second part deals with primitive transformation monoids and the third part studies modules associated with transformation monoids. -- The author gives a specialization (in this context) of Schützenberger's theory of unambiguous matrix monoids and generalizes Green's theory from the context of modules to transformation monoids. Further, the author introduces the notions of orbital and orbital digraphs for transformation monoids and characterizes primitivity in terms of connectedness of orbital digraphs. After that the author computes the projective cover of the transformation module associated with the transformation monoid over a field of characteristic zero in the case of a transitive transformation or partial transformation monoid. The paper ends with applications of Markov chains to the study of transformation monoids.