# Proportoids

arXiv: https://arxiv.org/abs/2210.01751 · HTML: https://mlsanity.com/paper/2210.01751/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 22 references that are in the map.
- First posted on arXiv 31 Aug 2022.

## Builds on (most cited first, 4 of 4)

- [Analogical proportions](https://mlsanity.com/paper/2006.02854/) (2020)
- [Logic program proportions](https://mlsanity.com/paper/1809.09938/) (2018)
- [Galois theory for analogical classifiers](https://mlsanity.com/paper/2205.04593/) (2022)
- [Analogical proportions in monounary algebras](https://mlsanity.com/paper/2208.06829/) (2022)

## Built on by (most cited first, 4 of 4)

- [Analogical proportions II](https://mlsanity.com/paper/2405.13461/) (2024)
- [Logic-based similarity](https://mlsanity.com/paper/2305.00065/) (2023)
- [Similarity-based analogical proportions](https://mlsanity.com/paper/2402.18360/) (2024)
- [Neural logic programs and neural nets](https://mlsanity.com/paper/2406.11888/) (2024)

## Abstract (arXiv)

Analogical proportions are expressions of the form ``a is to b what c is to d'' at the core of analogical reasoning, which itself is at the core of artificial intelligence. This paper contributes to the mathematical foundations of analogical proportions in the axiomatic tradition as initiated -- in the tradition of the ancient Greeks -- by Yves Lepage two decades ago. More precisely, we first introduce the name ``proportoid'' for sets endowed with a 4-ary analogical proportion relation satisfying a suitable set of axioms. We then study study different kinds of proportion-preserving mappings and relations and their properties. Formally, we define homomorphisms of proportoids as mappings mathsf H satisfying a:b::c:d iff mathsf Ha: mathsf Hb:: mathsf Hc: mathsf Hd for all elements and show that their kernel is a congruence. Moreover, we introduce (proportional) analogies as mappings mathsf A satisfying a:b:: mathsf Aa: mathsf Ab for all elements a and b in the source domain and show how to compute partial analogies. We then introduce a number of useful relations between functions (including homomorphisms and analogies) on proportoids and study their properties. In a broader sense, this paper is a further step towards a mathematical theory of analogical proportions.