# The matched product of set-theoretical solutions associated with shelves

## TL;DR

This paper studies the algebraic structure of solutions related to shelves, simplifying conditions for their matched products, and analyzing their properties, including non-degeneracy and independence from actions.

## Contribution

It provides simplified criteria for constructing matched products of solutions from shelves and analyzes their structural properties and invariance.

## Key findings

- Simplified conditions for matched product construction
- Criteria for left non-degeneracy of solutions
- Proved independence of structure shelf from action choices

## Abstract

We investigate the matched product of solutions associated with right and left shelves. First, we prove that the requirements to provide the matched product of solutions that come from shelves can be simplified. Then we give conditions for left non-degeneracy of the matched product. Later, we compute the structure shelf of the matched product of solutions. Finally, we prove that the structure shelf of the matched product does not depend on the choice of the actions.