Chains and anti-chains in the lattice of epigroup varieties

TL;DR

This paper investigates the structure of the lattice of epigroup varieties, demonstrating that certain intervals contain chains isomorphic to real numbers and anti-chains of continuum size, revealing complex order-theoretic properties.

Contribution

It proves that the intervals between specific epigroup varieties contain both chains and anti-chains of large cardinality, highlighting intricate lattice structures.

Findings

Intervals contain chains isomorphic to real numbers

Intervals contain anti-chains of continuum cardinality

Lattice of epigroup varieties has complex order-theoretic features

Abstract

Let $\mathcal E_n$ be the variety of all epigroups of index $\le n$. We prove that, for an arbitrary natural number $n$, the interval $[\mathcal E_n, \mathcal E_{n+1}]$ of the lattice of epigroup varieties contains a chain isomorphic to the chain of real numbers with the usual order and an anti-chain of the cardinality continuum.