On absolute nilpotent and idempotent elements of an evolution algebra corresponding to permutations

TL;DR

This paper characterizes absolute nilpotent and idempotent elements in evolution algebras linked to permutations, and decomposes these algebras into direct sums based on permutation cycles.

Contribution

It provides a detailed description of nilpotent and idempotent elements in permutation-based evolution algebras and introduces a decomposition method based on permutation cycles.

Findings

Explicit descriptions of nilpotent and idempotent elements for permutation-based evolution algebras

Decomposition of these algebras into direct sums according to permutation cycles

Enhanced understanding of algebra structure related to permutations

Abstract

We describe absolute nilpotent and some idempotent elements of an $n$- dimensional evolution algebra corresponding to two permutations and we decompose such algebras to the direct sum of evolution algebras corresponding to cycles of the permutations.