19.18 (2018)

Given an infinite set $\Omega$, define an algebra $A$ (the reduced incidence algebra of finite subsets) as follows. Let $V_n$ be the set of functions from the set of $n$-element subsets of $\Omega$ to the rationals $\mathbb{Q}$. Now let $A = \bigoplus V_n$, with multiplication as follows: for $f \in V_n$, $g \in V_m$, and $|X| = m + n$, let $(fg)(X) = \sum f(Y)g(X \setminus Y)$, where the sum is over the $n$-element subsets $Y$ of $X$. If $G$ is a permutation group on $\Omega$, let $A^G$ be the algebra of $G$-invariants in $A$.

Suppose that $G$ has no finite orbits on $\Omega$. It is known that then $A^G$ is an integral domain (M. Pouzet, Theor. Inf. App., 42, no. 1 (2008), 83–103). Is it true that the quotient of $A^G$ by the ideal generated by constant functions is also an integral domain?