Binary perm algebras and alternative algebras

TL;DR

This paper explores the structure of binary perm algebras, a subvariety of alternative algebras, by identifying their defining identities, constructing a basis, and classifying identities under the commutator operation.

Contribution

It introduces the defining identities of binary perm algebras, constructs their free algebra basis, and classifies identities related to the commutator, advancing understanding of this algebraic variety.

Findings

Identified defining identities of binary perm algebras.

Constructed a basis for free binary perm algebras.

Classified identities satisfied under the commutator.

Abstract

In this paper, we describe the defining identities of a variety of binary perm algebras, which is a subvariety of the variety of alternative algebras. In addition, we construct a basis of the free binary perm algebra and find a complete list of identities which satisfy binary perm algebra under commutator.