Partial magmatic bialgebras

TL;DR

This paper introduces partial magmatic bialgebras, a new algebraic structure with specific operations and co-operations, and proves an analogue of the Poincaré-Birkhoff-Witt theorem for them.

Contribution

It defines partial magmatic bialgebras and establishes a PBW-type theorem, extending classical algebraic results to this new framework.

Findings

Defined partial magmatic bialgebras with specific operations and co-operations.

Proved an analogue of the Poincaré-Birkhoff-Witt theorem for these structures.

Extended classical algebraic theorems to new algebraic frameworks.

Abstract

A partial magmatic bialgebra, (T;S)-magmatic bialgebra where T \subset S are subsets of the set of positive integers, is a vector space endowed with an n-ary operation for each n in S and an m-ary co-operation for each m in T satisfying some compatibility and unitary relations. We prove an analogue of the Poincar\'e-Birkhoff-Witt theorem for these partial magmatic bialgebras.