Largest subsemigroups of the full transformation monoid

In this paper we are concerned with the following question: for a semigroup S  , what is the largest size of a subsemigroup T⩽ST⩽S where T has a given property? The semigroups S that we consider are the full transformation semigroups; all mappings from a finite set to itself under composition of mappings. The subsemigroups T   that we consider are of one of the following types: left zero, right zero, completely simple, or inverse. Furthermore, we find the largest size of such subsemigroups UU where the least rank of an element in UU is specified. Numerous examples are given.