Pierce stalks in preprimal varieties

TL;DR

This paper characterizes the Pierce stalks of preprimal varieties associated with certain maximal clones, completing the description for cases involving central and equivalence relations.

Contribution

It provides a description of Pierce stalks for preprimal varieties related to central and non-trivial equivalence relations, extending previous work.

Findings

Pierce stalks are described for cases 6 and 7.

Utilizes central element theory for analysis.

Completes the classification for all seven types.

Abstract

An algebra $\mathbf{P}$ is called \textit{preprimal} if $\mathbf{P}$ is finite and $\func{Clo}(\mathbf{P})$ is a maximal clone. A \textit{preprimal variety} is a variety generated by a preprimal algebra. After Rosenberg's classification of maximal clones \cite{ro}; we have that a finite algebra is preprimal if and only if its term operations are exactly the functions preserving a relation of one of the following seven types: 1. Permutations with cycles all the same prime length, 2. Proper subsets, 3 Prime-affine relations, 4. Bounded partial orders, 5. $h$-adic relations, 6. Central relations $h\geq 2$, 7. Proper, non-trivial equivalence relations. In \cite{kn} Knoebel studies the Pierce sheaf of the different preprimal varieties and he asks for a description of the Pierce stalks. He solves this problem for the cases 1.,2. and 3. and left open the remaining cases. In this paper, using central element theory we succeeded in describing the Pierce stalks of the cases 6. and 7..