Varieties generated by compact metric spaces (Q2910256)

Monads (or triples) and their algebras are frequently related with (not necessarily finitary) universal algebra [the author, in: Handbook of algebra. Volume 3. Amsterdam: Elsevier. 67--153 (2003; Zbl 1064.18003)]. For instance, in [\textit{J. Isbell}, Algebra Univers. 15, 153--155 (1982; Zbl 0516.18008)], it is established that the category of commutative \(\mathbb{C}^\ast\)-algebras is a certain variety of algebras in a signature with a few finitary operations and a single \(\omega\)-operation.NEWLINENEWLINEThe present paper establish results on universal algebra aspects of the category of compact metrizable spaces: the convenient signature \(\Sigma\) contains only an \(\omega\)-operation. The main category-theoretic notion employed in the paper is that of the submonads of the monad \(\beta\) that to each discrete space associates its Stone-Čech compactification given by the set of all ultrafilters on each given set (= discrete space). Let \(r \in \beta(\omega) \setminus \omega\): (i) if \(X\) is a compact metrizable space, then \(X\) becomes a \(\Sigma\)-algebra in the following way: For each \(f \in X^{\omega}\) associate the unique point \(x \in X\) such that the ultrafilter \(\beta(f)(r)\) converges to \(x\); (ii) consider \({\mathcal V}_r\), the variety generated by all the \(\Sigma\)-algebras determined by compact metrizable spaces \(X\). The main results of this work are: (i) \({\mathcal V}_r\) can be identified with a certain full subcategory of the category of countably tight spaces; (ii) the set of equations satisfied by the \(\Sigma\)-algebra \((2,\chi_r)\) determines the variety \({\mathcal V}_r\), where \(\chi_r : 2^\omega \rightarrow 2\) is the characteristic function of \(r \subseteq 2^\omega\).