  The aim of this paper is to develop a theory of finite transformation monoids
and in particular to study primitive transformation monoids. We introduce the
notion of orbitals and orbital digraphs for transformation monoids and prove a
monoid version of D. Higman's celebrated theorem characterizing primitivity in
terms of connectedness of orbital digraphs. A thorough study of the module (or
representation) associated to a transformation monoid is initiated. In
particular, we compute the projective cover of the transformation module over a
field of characteristic zero in the case of a transitive transformation or
partial transformation monoid. Applications of probability theory and Markov
chains to transformation monoids are also considered and an ergodic theorem is
proved in this context. In particular, we obtain a generalization of a lemma of
P. Neumann, from the theory of synchronizing groups, concerning the partition
associated to a transformation of minimal rank.
