Standard Polynomials for Principal Subalgebras $\mathbb{K}{Q}_{\ge 1}$ of Path Algebras

We investigate standard polynomials for principal subalgebras of path algebras. First, we use standard polynomials to study the $PI$-theory of principal subalgebras. Then we describe the $S{t}_2$-elements and $S{t}_3$-elements of principal subalgebras, giving a characterization of their centers and 3-centers. In addition, we apply these results to combinatorics on words of formal languages, obtaining some explanations from a combinatorial perspective.