On a Question of Gr\"atzer and Lakser from the 1971 {\sl Transactions of the American Mathematical Society}

TL;DR

This paper addresses a question from 1971 about characterizing certain pseudocomplemented distributive lattices through homomorphism properties, providing a definitive answer to an open problem in lattice theory.

Contribution

It provides a complete characterization of pseudocomplemented distributive lattices in the specified amalgamation class, resolving a longstanding open question from 1971.

Findings

Characterization of pseudocomplemented distributive lattices without certain homomorphisms.

Resolution of the 1971 question posed by Grätzer and Lakser.

New criteria for lattice homomorphism properties in the specified class.

Abstract

Gr\"atzer and Lakser asked in the 1971 {\sl Transactions of the American Mathematical Society} if the pseudocomplemented distributive lattices in the amalgamation class of the subvariety generated by ${\bf 2}^n\oplus{\bf 1}$ can be characterized by the property of not having a $*$-homomorphism onto ${\bf 2}^i\oplus{\bf 1}$ for $1<i<n$. In this article, this question is answered. If you want to know the answer, you will have to read it (or skip to the last section).