On the networks of large embeddings

TL;DR

This paper introduces a network-based framework to measure the dissimilarity between large algebraic embeddings across various algebraic classes, providing a new perspective on algebraic similarity and diversity.

Contribution

It defines a novel network model for large embeddings and a corresponding distance measure applicable to different algebraic structures, including examples from group theory and monounary algebras.

Findings

The network captures large embeddings in algebraic classes.

The distance can be finite or infinite, indicating degrees of dissimilarity.

Examples demonstrate the framework's applicability across algebraic areas.

Abstract

We define a special network that exhibits the large embeddings in any class of similar algebras. With the aid of this network, we introduce a notion of distance that conceivably counts the minimum number of dissimilarities, in a sense, between two given algebras in the class in hand; with the possibility that this distance may take the value $\infty$. We display a number of inspirational examples from different areas of algebra, e.g., group theory and monounary algebras, to show that this research direction can be quite remarkable.