Evolution algebras generated by Gibbs measures

TL;DR

This paper explores the algebraic structures of function spaces derived from graphs and Gibbs measures, providing a constructive approach and structural theorems for finite and infinite graphs, with applications in various scientific fields.

Contribution

It introduces a method to associate evolution algebras with function spaces defined by graphs and Gibbs measures, including a structure theorem for finite connected graphs and insights into infinite graph cases.

Findings

Finite graphs have unique algebraic structures with isomorphic evolution algebras.

Constructed evolution algebras reveal properties useful for modeling biological, physical, and mathematical systems.

Infinite graphs enable the integration of thermodynamics into the study of evolution algebras.

Abstract

In this article we study algebraic structures of function spaces defined by graphs and state spaces equipped with Gibbs measures by associating evolution algebras. We give a constructive description of associating evolution algebras to the function spaces (cell spaces) defined by graphs and state spaces and Gibbs measure $\mu$. For finite graphs we find some evolution subalgebras and other useful properties of the algebras. We obtain a structure theorem for evolution algebras when graphs are finite and connected. We prove that for a fixed finite graph, the function spaces has a unique algebraic structure since all evolution algebras are isomorphic to each other for whichever Gibbs measures assigned. When graphs are infinite graphs then our construction allows a natural introduction of thermodynamics in studying of several systems of biology, physics and mathematics by theory of evolution algebras.