Polynomially rich algebras (Q5929285)

An algebra \(A\) is polynomially rich if every mapping \(f: A^n\to A\) that preserves congruences and their labeling (in the sense of tame congruence theory) is a polynomial of \(A\). The authors show that a locally finite variety \(V\) in which all finite algebras are polynomially rich is congruence modular and every finite SI-algebra \(D\) of \(V\) satisfies two conditions:NEWLINENEWLINENEWLINE(1) the centralizer of the monolith \(\mu\) of \(D\) is not bigger than \(\mu\),NEWLINENEWLINENEWLINE(2) if \(\mu\) is Abelian then \((0,\mu)\)-minimal sets are polynomially equivalent to one-dimensional vector spaces over \(\text{GF}(p)\) for \(p\) prime.NEWLINENEWLINENEWLINEMoreover, if \(V\) is congruence permutable then \(V\) is polynomially rich if and only if every finite SI-algebra of \(V\) satisfies (1) and (2).