Congruences of maximum regular subsemigroups of variants of finite full transformation semigroups

Let T_X be the full transformation monoid over a finite set X, and fix some a∈T_X of rank r. The variant T_X^a has underlying set T_X, and operation f⋆g=fag. We study the congruences of the subsemigroup P=Reg(T_X^a) consisting of all regular elements of T_X^a, and the lattice Cong(P) of all such congruences. Our main structure theorem ultimately decomposes Cong(P) as a specific subdirect product of Cong(T_r), and the full equivalence relation lattices of certain combinatorial systems of subsets and partitions. We use this to give an explicit classification of the congruences themselves, and we also give a formula for the height of the lattice.