Fine decompositions of algebraic systems induced by bases

ABSTRACT

We consider algebraic systems with several products, including unary products, $\mathfrak{S}$ (as examples we can take linear spaces, algebras, superalgebras, hom-algebras, triple systems, hom-triple systems, Poisson algebras, Bol algebras, n-algebras, etc.) We show that any basis of $\mathfrak{S}$ gives rise to a decomposition of $\mathfrak{S}$ as a direct sum of indecomposable well-described ideals (fine decomposition). The simplicity of the components in this decomposition is also characterized. There are as many non-isomorphic fine decompositions as orbits in a determined action.

Keywords:

• Linear space
• algebra
• hom-algebra
• n-algebra
• algebraic system
• fine decomposition
• ideal
• structure theory

2020 MSCs:

• 15A21
• 15A69
• 17A01
• 17A60
• 17D30

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is supported by the PCI of the UCA ‘Teoría de Lie y Teoría de Espacios de Banach’, the PAI with project number FQM298 and by the project FEDER-Andalucia with number sol-201800107643.