The Fraser-Horn and Apple properties

Abstract:

We consider varieties $\mathcal {V}$ in which finite direct products are skew-free and in which the congruence lattices of finite directly indecomposables have a unique coatom. We associate with $\mathcal {V}$ a family of derived varieties, $d(\mathcal {V})$: a variety in $d(\mathcal {V})$ is generated by algebras ${\mathbf {A}}$ where the universe of ${\mathbf {A}}$ consists of a congruence class of the coatomic congruence of a finite directly indecomposable algebra ${\mathbf {B}} \in \mathcal {V}$ and the operations of ${\mathbf {A}}$ are those of ${\mathbf {B}}$ that preserve this congruence class. We also consider the prime variety of $\mathcal {V}$, denoted ${\mathcal {V}_0}$, generated by all finite simple algebras in $\mathcal {V}$. We show how the structure of finite algebras in $\mathcal {V}$ is determined to a considerable extent by ${\mathcal {V}_0}$ and $d(\mathcal {V})$. In particular, the free $\mathcal {V}$-algebra on $n$ generators, ${{\mathbf {F}}_\mathcal {V}}(n)$, has as many directly indecomposable factors as ${{\mathbf {F}}_{{\mathcal {V}_0}}}(n)$ and the structure of these factors is determined by the varieties $d(\mathcal {V})$. This allows us to produce in many cases explicit formulas for the cardinality of ${{\mathbf {F}}_\mathcal {V}}(n)$. Our work generalizes the structure theory of discriminator varieties and, more generally, that of arithmetical semisimple varieties. The paper contains many examples of algebraic systems that have been investigated in different contexts; we show how these all fit into a general scheme.