Equationally extremal semilattices

TL;DR

This paper investigates extremal properties of semilattices, identifying those with maximal or minimal numbers of consistent equations and solutions within classes of fixed order.

Contribution

It introduces the concept of extremal semilattices based on their equational properties and characterizes those with extremal numbers of solutions and equations.

Findings

Identifies semilattices with maximal number of consistent equations.

Finds semilattices with minimal number of consistent equations.

Determines semilattices with maximal sum of solutions to equations.

Abstract

In the current paper we study extremal semilattices with respect to their equational properties. In the class $\mathbf{S}_n$ of all semilattices of order $n$ we find semilattices which have maximal (minimal) number of consistent equations. Moreover, we find a semilattice in $\mathbf{S}_n$ with maximal sum of numbers of solutions of equations.