Standard Polynomials for Principal Subalgebras mathbb{K}Q_(geq 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 $St_2$-elements and $St_3$-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.