Absolutely \(f\)-equationally compact monoids. (Q2502258)

An algebra \(A\) is called \(f\)-equationally compact (1-equationally compact) if every finitely solvable system of equations of \(A\) containing altogether a finite number of variables (all containing one and the same variable, respectively) is solvable. In this paper the author considers \(f\)-equationally compactness in the category of left \(S\)-acts. A monoid \(S\) is called left absolutely \(f\)-equationally compact (left absolutely 1-equationally compact) if all left \(S\)-acts are \(f\)-equationally compact (respectively, 1-equationally compact). A characterization of a left absolutely \(f\)-equationally compact monoid, which is proved to be equivalent to a left absolutely 1-equationally compact monoid, is obtained. For a congruence \(\rho\) on an act \(A\), and \(a\in A\), the left congruence \(\rho^a\) on \(S\) is defined by \(u\rho^a v\Leftrightarrow ua\rho va\). It is shown that a monoid \(S\) is left absolutely \(f\)-equationally compact if and only if all left ideals of \(S\) are finitely generated and for every left congruence \(\rho\) of \(S\) and every finite set \(\{s_1,\dots,s_k\}\subseteq S\) there exist an element \(u\in S\) and a finitely generated subcongruence \(\psi\subseteq\rho\) such that \((s_i,s_iu)\in\rho\) for every \(i\) and \(\rho\subseteq\psi^u\cup(\rho^u|_J)\), where \(J\) is the left ideal of \(S\) generated by the set \(\{s_1,\dots,s_k\}\). In addition, it is got that every commutative absolutely 1-pure absolutely 1-equationally compact monoid is absolutely injective.