Construction of free commutative Reynolds algebras by Grobner-Shirshov bases

Reynolds algebras originated from the celebrated work of Reynolds in 1895 on turbulence theory in fluid mechanics. The subject has attracted broad interests in recent years. This paper constructs free commutative Reynolds algebras, responding to a problem posed by G. Birkhoff in 1961, by applying the method of Grobner-Shirshov bases. First presenting a free commutative Reynolds algebra as a quotient of the free commutative operated algebra, we then provide a linear basis of the quotient by establishing the composition-diamond lemma for commutative operated algebra and showing that the operated ideal generated by the Reynolds identity possesses a Grobner-Shirshov basis.(c) 2023 Elsevier Ltd. All rights reserved.

Automated summary

This paper constructs free commutative Reynolds algebras, responding to a problem posed by G. Birkhoff in 1961, by applying the method of Grobner-Shirshov bases.