Finite algebras of finite complexity

Abstract

We develop a new centrality concept and apply it to solve certain outstanding problems about ÿnite algebras. In particular, we describe all ÿnite algebras of ÿnite complexity and all ÿnite strongly abelian algebras which generate residually small varieties.

Residual Smallness and Weak Centrality

“…The basic observation is Lemma 4.4, which has a consequence of independent interest stated in Corollary 4.5: in a weakly abelian variety, every strongly nilpotent tolerance of every finite algebra is rectangular. Combined with the results of [5] this corollary implies that the weak abelian property is equivalent to residual smallness for strongly nilpotent varieties (see Corollary 4.6). The main application of Lemma 4.4 is Corollary 4.9.…”

“…An affirmative answer to a variant of this question has been given in [5]. In that paper the strong abelian property (also discovered by McKenzie) is considered instead of the "normal" version.…”

“…The nilpotence concept stemming from strong abelianness is called strong nilpotence. In [5] it is shown that strongly nilpotent algebras satisfy the RS-conjecture. It is also shown that the variety generated by a finite strongly nilpotent algebra is residually small if and only if every member of this variety is "abelian", but this is neither "normal", nor strong abelianness, but, rather, a new concept called rectangular abelianness, or simply rectangularity.…”

“…It is based on two constructions of large subdirectly irreducible algebras. One of these constructions is inherited from [5], the other one is new, but its origins are in the proof of the Freese-McKenzie Theorem. To prove the second main result we first have to establish some properties of weakly abelian varieties, this is done in Section 4.…”

An overview of modern universal algebra

“…Kearnes and Kiss in [18] introduced a strong centrality relation which captures strong abelianness, and teased from it two further notions of centrality (weak and rectangular ), which in turn give rise to their own notions of abelianness. As well, they solve the RS problems for finite strongly nilpotent algebras: Theorem 6.18.…”