### Abstract

In recent years, algebraic studies of the differential calculus and integral calculus in the forms of differential algebra and Rota-Baxter algebra have been merged together to reflect the close relationship between the two calculi through the First Fundamental Theorem of Calculus. In this paper we study this relationship from a categorical point of view in the context of distributive laws which can be tracked back to the distributive law of multiplication over addition. The monad giving Rota-Baxter algebras and the comonad giving differential algebras are constructed. Then we obtain monads and comonads giving the composite structures of differential and Rota-Baxter algebras. As a consequence, a mixed distributive law of the monad giving Rota-Baxter algebras over the comonad giving differential algebras is established.

Original language | English (US) |
---|---|

Pages (from-to) | 139-165 |

Number of pages | 27 |

Journal | Advances in Applied Mathematics |

Volume | 72 |

DOIs | |

State | Published - Jan 2016 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Applied Mathematics

### Keywords

- 12H05
- 13Nxx
- 16W99
- MSC primary 18C15
- secondary 18C20

### Cite this

}

*Advances in Applied Mathematics*, vol. 72, pp. 139-165. https://doi.org/10.1016/j.aam.2015.09.014

**Monads and distributive laws for Rota-Baxter and differential algebras.** / Zhang, Shilong; Guo, Li; Keigher, William.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Monads and distributive laws for Rota-Baxter and differential algebras

AU - Zhang, Shilong

AU - Guo, Li

AU - Keigher, William

PY - 2016/1

Y1 - 2016/1

N2 - In recent years, algebraic studies of the differential calculus and integral calculus in the forms of differential algebra and Rota-Baxter algebra have been merged together to reflect the close relationship between the two calculi through the First Fundamental Theorem of Calculus. In this paper we study this relationship from a categorical point of view in the context of distributive laws which can be tracked back to the distributive law of multiplication over addition. The monad giving Rota-Baxter algebras and the comonad giving differential algebras are constructed. Then we obtain monads and comonads giving the composite structures of differential and Rota-Baxter algebras. As a consequence, a mixed distributive law of the monad giving Rota-Baxter algebras over the comonad giving differential algebras is established.

AB - In recent years, algebraic studies of the differential calculus and integral calculus in the forms of differential algebra and Rota-Baxter algebra have been merged together to reflect the close relationship between the two calculi through the First Fundamental Theorem of Calculus. In this paper we study this relationship from a categorical point of view in the context of distributive laws which can be tracked back to the distributive law of multiplication over addition. The monad giving Rota-Baxter algebras and the comonad giving differential algebras are constructed. Then we obtain monads and comonads giving the composite structures of differential and Rota-Baxter algebras. As a consequence, a mixed distributive law of the monad giving Rota-Baxter algebras over the comonad giving differential algebras is established.

KW - 12H05

KW - 13Nxx

KW - 16W99

KW - MSC primary 18C15

KW - secondary 18C20

UR - http://www.scopus.com/inward/record.url?scp=84955140328&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84955140328&partnerID=8YFLogxK

U2 - 10.1016/j.aam.2015.09.014

DO - 10.1016/j.aam.2015.09.014

M3 - Article

AN - SCOPUS:84955140328

VL - 72

SP - 139

EP - 165

JO - Advances in Applied Mathematics

JF - Advances in Applied Mathematics

SN - 0196-8858

ER -