Residual Smallness and Weak Centrality

Abstract

Abstract. We develop a method of creating skew congruences on subpowers of finite algebras using groups of twin polynomials, and apply it to the investigation of residually small varieties generated by nilpotent algebras. We prove that a residually small variety generated by a finite nilpotent (in particular, a solvable Eminimal) algebra is weakly abelian. Conversely, we show in two special cases that a weakly abelian variety is residually bounded by a finite number: when it is generated by an E-minimal, or by a finite strongly nilpotent algebra. This establishes the RS-conjecture for E-minimal algebras.

An overview of modern universal algebra

“…In [19] the authors define the notion of the twin group of a (0, α)-trace and relate its properties to weak abelianness and nilpotence. In particular, they prove that if A is a finite nilpotent algebra, then a necessary consequence of HSP(A) being residually small is that the twin groups which arise must be abelian and hence every member of HSP(A) must be weakly abelian.…”