Generalized anti-commutative Gröbner-Shirshov basis theory and free Sabinin algebras

Abstract

E. Chibrikov defined regular monomials (here called Chibrikov words) and proved that they form a linear basis of a free Sabinin algebra. In this paper, we introduce the notion of a generalized anti-commutative algebra and establish Gröbner-Shirshov basis theory for those algebras. We provide another approach to the definition of Chibrikov words i.e. we find a generalized anti-commutative Gröbner-Shirshov S of a free Sabinin algebra such that Irr(S) is the set of all Chibrikov words on X, where Irr(S) is the set of all normal Ω-words not containing maximal monomials of polynomials from S. Following from Gröbner-Shirshov basis theory of generalized anti-commutative algebras, the set Irr(S) is a linear basis of a free Sabinin algebra.

Keywords:

• Generalized anti-commutative algebra
• Gröbner-Shirshov basis
• Sabinin algebra

2020 AMS Mathematics Subject Classification:

• 17A61
• 16S15
• 17A30

Acknowledgment

The authors would like to thank the anonymous referee for helpful comments.

Additional information

Funding

This work was financially supported by the NNSF of China (No. 11501237), the NSF of Guangdong Province (No. 2016A030310099), and the Huizhou University (hzuxl201523, hzu201704, hzu201804).