The model companion of the class of pseudocomplemented semilattices is finitely axiomatizable

TL;DR

This paper proves that the class of existentially closed pseudocomplemented semilattices has a finite axiomatization, providing an answer to a longstanding open problem about their model companion.

Contribution

It extends the finite axiomatization of algebraically closed pseudocomplemented semilattices to the existentially closed case, establishing the finite axiomatizability of their model companion.

Findings

The class of existentially closed pseudocomplemented semilattices is finitely axiomatizable.

This class coincides with the model companion of pseudocomplemented semilattices.

The paper solves the open problem posed by Albert and Burris.

Abstract

In this paper it is shown that the class $\mathcal{PCSL}^{ec}$ of existentially closed pseudocomplemented semilattices is finitely axiomatizable by appropriately extending the finite axiomatization of the class $\mathcal{\mathcal{PCSL}}^{ac}$ of algebraically closed pseudocomplemented semilattices presented in Rupp, Addler, and Schmid. Because $\mathcal{PCSL}^{ec}$ coincides with the model companion of the class $\mathcal{PCSL}$ of pseudocomplemented semilattices this addendum to Rupp, Addler, and Schmid solves the problem posed by Albert and Burris in the final paragraph of their paper: "Does the class of pseudocomplemented semilattices have a finitely axiomatizable model companion?"