On a generalized endomorphism (Q2911200)

An algebra \((A,F)\) is called entropic (or medial) if it satisfies the identity NEWLINE\[NEWLINE\begin{aligned} & g(f(x_{11},\dots ,x_{1n}), \dots ,f(x_{m1}, \dots ,x_{mn})) =\\ & f(g(x_{11}, \dots ,x_{m1}), \dots ,g(x_{1n},\dots x_{mn}))\end{aligned} NEWLINE\]NEWLINE for every \(n\)-ary \(f\) and \(m\)-ary \(g\) in \(F\).NEWLINENEWLINEA mapping \(f: A^n \to A\) of an algebra \((A,F)\) is called a generalized endomorphism if for each \(m\)-ary basic operation \(g\) there are \(n\)-ary term operations \(t_1, \dots , t_m\) such that NEWLINE\[NEWLINE\begin{aligned} & f(g(x_{11}, \dots ,x_{m1}), \dots ,g(x_{1n},\dots x_{mn})) = \\ & g(t_1(x_{11},\dots ,x_{1n}), \dots ,t_m(x_{m1}, \dots ,x_{mn})).\end{aligned}\tag{2}NEWLINE\]NEWLINENEWLINENEWLINEThe author provides several conditions under which an algebra having a generalized endomorphism has mutually entropic operations.NEWLINENEWLINEFor related results see also [\textit{J. Ježek} and \textit{T. Kepka}, ``Medial groupoids'', Rozpr. Cesk. Akad. Ved, Rada Mat. Prir. Ved 93, No. 2, 93 p. (1983; Zbl 0527.20044); \textit{A. B. Romanowska} and \textit{J. D. H. Smith}, Modes. Singapore: World Scientific (2002; Zbl 1012.08001), Chapter 5], and references therein.NEWLINENEWLINELet us note that the property defined by the second formula was introduced by \textit{A. Pilitowska} [Modes of submodes. Warsaw: Warsaw University of Technology (PhD Thesis) (1996)] under the name ``complex condition'', and later by \textit{K. Adaricheva}, \textit{A. Pilitowska} and \textit{D. Stanovský} [Algebra Logic 47, No. 6, 367--383 (2008); translation from Algebra Logika 47, No. 6, 655--686 (2008; Zbl 1241.08002)] under the name ``generalized entropic property''.