A new representation of finite Hoops using a new type of product of structures

TL;DR

This paper introduces a novel product construction for finite hoops, demonstrating that any finite hoop can be represented as a product of its filters and homomorphic images, with implications for their structure.

Contribution

It defines a new type of product for finite hoops and proves that all finite hoops can be decomposed into products of finite MV-chains using this construction.

Findings

Finite hoops can be represented as products of filters and homomorphic images.

The new product satisfies a form of associativity.

Every finite hoop is a product of finite MV-chains.

Abstract

In this paper we show that a new type of products hoops can be defined which, in the case of finite hoops, can describe an arbitrary hoop $\mathbf A$ as the product of its arbitrary filter $F$ and the corresponding homomorphic image $\mathbf A/F$. Moreover, this product satisfies a certain kind of associativity, and as a consequence we show that every finite hoop is in this sense a product of finite MV-chains.