A note on maximal regular subsemigroups of the finite transformation semigroups T (n, r)

摘要

Let T_n be the semigroup of all full transformations on the finite set X_n = {1, 2,..., n}. For 1 ≤ r ≤ n, set T (n, r) = {α ∈ T_n|rank(α) ≤ r}. In this note we show that, for 2 ≤ r ≤ n ? 2, any maximal regular subsemigroup of the semigroup T (n, r) is idempotent generated, but this may not happen in the semigroup T (n, n ?1).