The order of principal congruences of a semimodular lattice

TL;DR

This paper proves that any finite bounded ordered set can be represented as the order of principal congruences of a finite semimodular lattice, advancing understanding of lattice congruence structures.

Contribution

It establishes a representation theorem linking finite bounded ordered sets to principal congruences of finite semimodular lattices.

Findings

Finite bounded ordered sets can be represented as principal congruences of finite semimodular lattices.

Provides a constructive approach for such representations.

Enhances the theoretical framework of lattice congruences.

Abstract

This paper has been withdrawn by the authors due to a crucial computational error. In this paper we deal with the finite case. We prove that a finite bounded ordered set can be represented as the order of principal congruences of a finite \emph{semimodular lattice}.