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 = f a g. 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. [ABSTRACT FROM AUTHOR]