Equivariant algebraic and semi-algebraic geometry of infinite affine space

We study $\textrm{Sym}(\infty)$-orbit closures of not necessarily closed points in the Zariski spectrum of the infinite polynomial ring $\mathbb{C}[x_{ij}:\, i\in\mathbb{N},\,j\in[n]]$. Among others, we characterize invariant prime ideals in this ring. Furthermore, we study projections of basic equivariant semi-algebraic sets defined by $\textrm{Sym}(\infty)$ orbits of polynomials in $\mathbb{R}[x_{ij}:\, i\in\mathbb{N},\,j\in[n]]$. For $n=1$ we prove a quantifier elimination type result which fails for $n>1$.