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 $a\in T_X$ of rank r. The variant $T_X^a$ has underlying set $T_X$, and operation $f\star g=fag$. We study the congruences of the subsemigroup $P=\mathrm{Reg}\left(T_X^a\right)$ consisting of all regular elements of $T_X^a$, and the lattice $\mathrm{Cong}\left(P\right)$ of all such congruences. Our main structure theorem ultimately decomposes $\mathrm{Cong}\left(P\right)$ as a specific subdirect product of $\mathrm{Cong}\left(T_r\right)$, 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.