Congruences of regular variants of finite full transformation semigroups

Let TX be the full transformation monoid over a finite set X, and fix some a∈ TX of rank r. The variant TXᵃ has underlying set TX, and operation f g=fag. We study the congruences of the subsemigroup P=Reg(TXᵃ) consisting of all regular elements of TXᵃ, and the lattice $Cong(P)$ of all such congruences. Our main structure theorem ultimately decomposes $Cong(P)$ as a specific subdirect product of Cong(Tᵣ) 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.