The strong universality of ANRs with a suitable algebraic structure

TL;DR

This paper establishes a universal property for ANR spaces with algebraic operations, linking topological and algebraic structures, and applies it to Lawson semilattices, broadening understanding of their universality.

Contribution

It provides a new characterization of strong universality for ANRs with algebraic operations, connecting topological classes with algebraic structures and applying it to Lawson semilattices.

Findings

Characterization of strong universality in ANRs with algebraic operations.

Identification of conditions for universal mappings involving compact spaces.

Application to detecting strongly universal Lawson semilattices.

Abstract

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 $2^\omega$-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 $(\mathbf \Pi^0_1\cap\mathcal C,\mathcal C)$-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.