Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Michal Botur

TL;DR

The paper introduces an $f$-product for hoops, a wreath-like associative product that combines a filter $F$ with the quotient $A/F$ into a single hoop via a product morphism $f$. It proves that finite hoops admit a canonical representation $A \cong F \ltimes_f (A/F)$ and, furthermore, decomposes every finite hoop into a finite sequence of MV-chains connected by $f$-products, with the count tied to the idempotent-chain length. An Appendix links these constructions to exact sequences, showing how composition of product morphisms corresponds to kernel-and-image data. The work provides a Krohn–Rhodes–inspired structural framework for finite residuated lattices, offering new avenues for understanding MV-algebras and their finite decompositions.

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.

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

TL;DR

The paper introduces an -product for hoops, a wreath-like associative product that combines a filter with the quotient into a single hoop via a product morphism . It proves that finite hoops admit a canonical representation and, furthermore, decomposes every finite hoop into a finite sequence of MV-chains connected by -products, with the count tied to the idempotent-chain length. An Appendix links these constructions to exact sequences, showing how composition of product morphisms corresponds to kernel-and-image data. The work provides a Krohn–Rhodes–inspired structural framework for finite residuated lattices, offering new avenues for understanding MV-algebras and their finite decompositions.

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 as the product of its arbitrary filter and the corresponding homomorphic image . 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.

Paper Structure

This paper contains 6 sections, 12 theorems, 82 equations, 5 figures.

Table of Contents

  1. Motivation
  2. Introduction
  3. $f$-product of hoops
  4. Decomposition of finite hoops
  5. Appendix : The relation of $f$-products to exact sequences
  6. Open problems

Key Result

Lemma 2

If $\mathbf A=(A;\cdot,\rightarrow,1)$ is a hoop and $x,y,z\in A$ then it satisfies

Figures (5)

  • Figure 1: Idea of a wreath product of semigroup actions.
  • Figure 2: Associativity of a wreath product.
  • Figure 3: The visualisation of $\mathbf M\ltimes_{\psi_f}\mathbf A.$
  • Figure 4: The visualisation of $\mathbf A\ltimes_{\psi_f}\mathbf M.$
  • Figure 5: The composition of exact sequences

Theorems & Definitions (30)

  • Definition 1
  • Lemma 2: Haj
  • Lemma 3
  • proof
  • Lemma 4: Galatos, Tsinakis
  • Lemma 5
  • proof
  • Definition 6
  • Definition 7
  • Theorem 8
  • ...and 20 more