Moves for Bergman algebras

TL;DR

This paper introduces Bergman presentations and algebras, visualized by Bergman graphs, and establishes moves that preserve their algebraic classes, generalizing known results from Leavitt path algebras.

Contribution

It defines moves on Bergman graphs that preserve algebraic isomorphism and Morita equivalence, extending classical graph moves to a broader algebraic context.

Findings

Moves preserve isomorphism classes of Bergman algebras.

Moves preserve Morita equivalence classes of Bergman algebras.

Connections between Tietze transformations and Bergman graph moves.

Abstract

We define Bergman presentations and Bergman algebras associated to Bergman presentations. These algebras embrace various generalisations of Leavitt path algebras. A Bergman presentation can be visualised by a Bergman graph, which is a finite bicoloured hypergraph satisfying two conditions. We define several moves for Bergman graphs and prove that they preserve the isomorphism class (respectively the Morita equivalence class) of the corresponding Bergman algebra. One recovers the well-known results, that in the context of finite directed graphs the shift move, outsplitting, insplitting, source elimination and collapsing preserve the isomorphism class (respectively the Morita equivalence class) of the corresponding Leavitt path algebra. Moreover, we mention some connections between Tietze transformations and the moves for Bergman graphs defined in this paper.