Cohomology of finite monogenic self-distributive structures

TL;DR

This paper computes the cohomology of finite monogenic shelves, including Laver tables and cyclic racks, and develops tools for studying shelf cohomology, highlighting the role of cyclic sub-racks.

Contribution

It introduces methods for calculating cohomology of finite monogenic shelves and identifies cyclic sub-racks as key structures within them.

Findings

Cohomology of FMS computed with arbitrary coefficients.

Development of general tools for shelf cohomology analysis.

Identification of cyclic sub-racks as characteristic substructures.

Abstract

A shelf is a set with a binary operation~$\op$ satisfying $a \op (b \op c) = (a \op b) \op (a \op c)$. Racks are shelves with invertible translations $b \mapsto a \op b$; many of their aspects, including cohomological, are better understood than those of general shelves. Finite monogenic shelves (FMS), of which Laver tables and cyclic racks are the most famous examples, form a remarkably rich family of structures and play an important role in set theory. We compute the cohomology of FMS with arbitrary coefficients. On the way we develop general tools for studying the cohomology of shelves. Moreover, inside any finite shelf we identify a sub-rack which inherits its major characteristics, including the cohomology. For FMS, these sub-racks are all cyclic.