Special elements of the lattice of monoid varieties

TL;DR

This paper classifies special elements within the lattice of monoid varieties, revealing their properties and relationships, and reduces complex problems to the case of completely regular varieties.

Contribution

It provides a complete classification of neutral and costandard elements and characterizes upper-modular elements in the lattice of monoid varieties.

Findings

All neutral and costandard elements are classified.

Upper-modular elements are either completely regular or commutative, except the variety of all monoids.

All commutative monoid varieties are codistributive elements.

Abstract

We completely classify all neutral or costandard elements in the lattice $\mathbb{MON}$ of all monoid varieties. Further, we prove that an arbitrary upper-modular element of $\mathbb{MON}$ except the variety of all monoids is either a completely regular or a commutative variety. Finally, we verify that all commutative varieties of monoids are codistributive elements of $\mathbb{MON}$. Thus, the problems of describing codistributive or upper-modular elements of $\mathbb{MON}$ are completely reduced to the completely regular case.