Lattices of congruence relations for inverse semigroups

The study of congruence relations is acknowledged as fundamental to the study of algebras. Inverse semigroups are a widely studied class for which congruences are well understood. We study one sided congruences on inverse semigroups. We develop the notion of an inverse kernel and show that a left congruence is determined by its trace and inverse kernel. Our strategy identifies the lattice of left congruences as a subset of the direct product of the lattice of congruences on the idempotents and the lattice of full inverse subsemigroups. This is a natural way to describe one sided congruences with many desirable properties, including that a pair is the inverse kernel and trace of a left congruence precisely when it is the inverse kernel and trace of a right congruence. We classify inverse semigroups for which every Rees left congruence is finitely generated, and provide alternative proofs to classical results, including classifications of left Noetherian inverse semigroups, and Clifford semigroups for which the lattice of left congruences is modular or distributive. In the second half of this thesis we study the partial automorphism monoid of a finite rank free group action. Congruences are described in terms a Rees congruence, subgroups of direct powers of the group and a subgroup of the wreath product of the group and a symmetric group. Via analysis of the subgroups arising in this description we show that, for finite groups, the number of congruences grows polynomially in the rank of action with an exponent related to the chief length of the group