Minimum degrees of finite rectangular bands, null semigroups, and variants of full transformation semigroups

For a positive integer n, the full transformation semigroup T_n consists of all self maps of the set {1,…,n} under composition. Any finite semigroup S embeds in some T_n, and the least such n is called the (minimum transformation) degree of S and denoted μ(S). We find degrees for various classes of finite semigroups, including rectangular bands, rectangular groups and null semigroups. The formulae we give involve natural parameters associated to integer compositions. Our results on rectangular bands answers a question of Easdown from 1992, and our approach utilises some results of independent interest concerning partitions/colourings of hypergraphs.

As an application, we prove some results on the degree of a variant T_n^a. (The variant S^a = (S,_*) of a semigroup S with respect to a fixed element a∈S, has underlying set S and operation x_*y = xay.) It has been previously shown that n ≤ μ(T_n^a) ≤ 2n−r if the sandwich element a has rank r, and the upper bound of 2n−r is known to be sharp if r ≥ n−1. Here we show that μ(T_n^a) = 2n−r for r ≥ n−6. In stark contrast to this, when r = 1, and the above inequality says n ≤ μ(T_n^a) ≤ 2n−1, we show that μ(T_n^a)/n → 1 and μ(T_n^a)−n → ∞ as n → ∞.

Among other results, we also classify the 3-nilpotent subsemigroups of T_n, and calculate the maximum size of such a subsemigroup.