Linear semigroups of polynomial growth in positive characteristic Original Research Article

Finitely generated subsemigroups S of the full matrix monoid Mn(K) over a field K of positive characteristic are considered. It is shown that S has polynomial growth if and only if every nonempty intersection S∩G with a maximal subgroup G of Mn(K) has a finitely generated nilpotent-by-finite group of fractions. Moreover, the degree of growth of such an S is bounded by a function f(n, r) of n and of the maximal rank r of a free commutative subsemigroup of S.