Tying up baric algebras

TL;DR

This paper introduces a new construction for combining two baric algebras into a single algebra, explores its properties, and classifies certain cases, enhancing understanding of algebraic structures.

Contribution

It defines a novel method to construct a combined baric algebra from two given ones and classifies associative, countable-dimensional cases in zero characteristic.

Findings

The construction preserves indecomposability in the commutative, unital case.

Properties of the new algebra structure are straightforward and well-behaved.

Classification of associative, countable-dimensional, zero-characteristic algebras of this form is achieved.

Abstract

Given two baric algebras $(A_1,\omega_1)$ and $(A_2,\omega_2)$ we describe a way to define a new baric algebra structure over the vector space $A_1\oplus A_2$, which we shall denote $(A_1\bowtie A_2,\omega_1\bowtie\omega_2)$. We present some easy properties of this construction and we show that in the commutative and unital case it preserves indecomposability. Algebras of the form $A_1\bowtie A_2$ in the associative, coutable-dimensional, zero-characteristic case are classified.