The ubiquity of free subsemigroups of infinite triangular matrices (Q1865682)

The author shows that almost all \(k\)-generator subsemigroups of the multiplicative semigroup of infinite triangular marices over a finite field, \(S:=T(\infty,p^r)\), are free of rank \(k\) in the sense that \(S^k\setminus F\) is a meager subset of \(S^k\), where \(F=\{x\in S^k\mid\langle x\rangle\) is a free subsemigroup of rank \(k\}\).