Strongly fluid and weakly unsolid varieties (Q2762665)

Let \(F=\{f_i: i\in I\}\) be a set of operations of type \(\tau, W_{\tau}(X)\) be the set of all terms over a set of varieties \(X\), built by \(F\). A mapping \(\sigma:F\Rightarrow W_{\tau}(X)\) is called a hypersubstition; it can be in a natural way extended to \(\widehat{\sigma}: W_{\tau}(X)\Rightarrow W_{\tau}(X)\). For an algebra \(\underline{A}=(A,F)\) and a hypersubstition \(\sigma\), let the derived algebra be defined as \(\sigma[\underline{A}]:=(A,\{\sigma(f_i):i\in I\})\). Let \(V\) be a variety of type \(\tau\) and \(\sigma_1,\sigma_2\) be hypersubstitions; the authors define \(\sigma_1 \sim^I_V \sigma_2:\Leftrightarrow (\forall \underline{A} \in V)(\sigma_1[\underline{A}]\cong \sigma_2[\underline{A}])\). In the paper, the following theorem is proved: Let \(V\) be a variety of type \(\tau, M\) be a monoid of hypersubstitions. Let \(\Phi\) be a choice function which chooses one hypersubstition from each class of the equivalence relation \(\sim^I_V \restriction M\). If the set of hyperidentities chosen in this way is finite and if \(V\) has a finite equational basis \(\Sigma\), the system of hyperidentities generated by the set \(\varGamma:=\{\widehat{\sigma}(s)\approx \widehat{\sigma}(t)): s\approx t\in \Sigma, \sigma \in M\}\) is also finitely based.