Dense semigroups of triangular matrices

TL;DR

This paper demonstrates that the set of all lower triangular matrices over real or complex numbers contains dense subsemigroups generated by only $n+1$ matrices, revealing a minimal generating set for density.

Contribution

It establishes that $T_n(K)$ has dense subsemigroups generated by $n+1$ matrices, providing new insights into the algebraic structure of triangular matrix semigroups.

Findings

Existence of dense subsemigroups in $T_n(K)$ generated by $n+1$ matrices

Minimal generating set size for density in triangular matrix semigroups

Extension of density results to lower triangular matrices over $R$ and $C$

Abstract

Let $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$, and $T_n(\mathbb{K})$ be the set of $n\times n$ lower triangular matrices with entries in $\mathbb{K}$. We show that $T_n(\mathbb{K})$ has dense subsemigroups that are generated by $n+1$ matrices.