Gröbner bases for operads

Implementing Gröbner bases for operads

We present an implementation of the algorithm for computing Gröbner bases for operads due to the first author and A. Khoroshkin. We discuss the actual algorithms, as well as the choices made for the implementation platform and the data representation. We indicate strengths and weaknesses of our approach, and discuss possible directions for expanding the current work.

Gröbner Bases for Operads

We define a new monoidal category on collections (shuffle composition). Monoids in this category (shuffle operads) turn out to bring a new insight in the theory of symmetric operads. For this category, we develop the machinery of Gröbner bases for operads, and present operadic versions of Bergman’s Diamond Lemma and Buchberger’s algorithm. This machinery can be applied to study symmetric operad...

Multiplicative Bases, Gröbner Bases, and Right Gröbner Bases

Before surveying the results of the paper, we introduce path algebras. Path algebras play a central role in the representation theory of finite-dimensional algebras (Gabriel, 1980; Auslander et al., 1995; Bardzell, 1997) and the theory of Gröbner bases (Bergman, 1978; Mora, 1986; Farkas et al., 1993) has been an important tool in some results (Feustel et al., 1993; Green and Huang, 1995; Bardze...

Gröbner-Shirshov bases for the non-symmetric operads of dendriform algebras and quadri-algebras

Gröbner Bases for Linearized Polynomials

In this work we develop the theory of Gröbner bases for modules over the ring of univariate linearized polynomials with coefficients from a finite field.