Largest subsemigroups of the full transformation monoid

AbstractIn this paper we are concerned with the following question: for a semigroup S, what is the largest size of a subsemigroup T⩽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 U where the least rank of an element in U is specified. Numerous examples are given