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 $*:Mtimes Mto M$ such that for every $x,yin M$ the inclusion $x*yin X$ holds if and only if $x,yin X$. Assume also that there exist continuous unary operations $u,v:Mto M$ such that $x=u(x)*v(x)$ for all $xin M$. Given a $2^omega$-stable $mathbf Pi^0_2$-hereditary weakly $mathbf Sigma^0_2$-additive class of spaces $mathcal C$, we prove that the pair $(M,X)$ is strongly $(mathbf Pi^0_1capmathcal C,mathcal C)$-universal if and only if for any compact space $Kinmathcal C$, subspace $Cinmathcal C$ of $K$ and nonempty open set $Usubseteq M$ there exists a continuous map $f:Kto U$ such that $f^{-1}[X]=C$. This characterization is applied to detecting strongly universal Lawson semilattices.