The strong universality of ANRs with a suitable algebraic structure

Let $M$ be an ANR space and $X$ be a homotopy dense subspace in $M$. Assume that $M$ admits a continuous binary operation $*:M\times M\to M$ such that for every $x,y\in M$ the inclusion $x*y\in X$ holds if and only if $x,y\in X$. Assume also that there exist continuous unary operations $u,v:M\to M$ such that $x=u(x)*v(x)$ for all $x\in M$. Given a $2^ω$-stable $\mathbf Π^0_2$-hereditary weakly $\mathbf Σ^0_2$-additive class of spaces $\mathcal C$, we prove that the pair $(M,X)$ is strongly $(\mathbf Π^0_1\cap\mathcal C,\mathcal C)$-universal if and only if for any compact space $K\in\mathcal C$, subspace $C\in\mathcal C$ of $K$ and nonempty open set $U\subseteq M$ there exists a continuous map $f:K\to U$ such that $f^{-1}[X]=C$. This characterization is applied to detecting strongly universal Lawson semilattices.

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