PPubMed4 Mar 2019Cited 4×

Classification of Rota-Baxter operators on semigroup algebras of order two and three S. Zheng et al. Communications in Algebra

Shanghua Zheng, Li Guo, Markus Rosenkranz

Abstract

In this paper, we determine all the Rota-Baxter operators of weight zero on semigroup algebras of order two and three with the help of computer algebra. We determine the matrices for these Rota-Baxter operators by directly solving the defining equations of the operators. We also produce a Mathematica procedure to predict and verify these solutions.

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All weight-zero Rota‑Baxter operators on semigroup algebras of order 2 and 3 are classified, with explicit matrix families and a Mathematica routine for their computation.

What’s new

i.Complete classification of weight‑zero Rota‑Baxter operators on semigroup algebras of order 2.
ii.Complete classification of weight‑zero Rota‑Baxter operators on semigroup algebras of order 3 (both commutative and non‑commutative cases).
iii.Development of a Mathematica procedure (FindRBO) to generate and verify the operator matrices.
iv.Presentation of explicit matrix families for each semigroup in detailed tables.

How they did it

The authors derived the defining equations for a weight‑zero Rota‑Baxter operator on a semigroup algebra, encoded these equations in Mathematica, and used the FindRBO function to solve them; the solutions were then checked by hand to provide rigorous proofs.

What they found

For order 2 semigroups, the matrices of all Rota‑Baxter operators are given in Table 4 (four distinct semigroups, with parameterized families such as (a − a b − b) and the zero matrix).
For the 12 commutative semigroups of order 3, the operator matrices are listed in Table 1 and Table 3 (e.g., C₁,₁ = (a b − a − b c d − c − d e f − e − f) with six free parameters).
For the 6 non‑commutative semigroups of order 3, the matrices are given in Table 2 and Table 6 (including families like N₁,₁ = (0 0 0 0 0 0 0 a 0) with a ≠ 0).

Caveats

✕Only weight‑zero Rota‑Baxter operators are treated.
✕The classification is limited to semigroup algebras of order 2 and 3; higher orders are not addressed.
✕Results rely on computer‑algebra assistance, which may not scale easily to larger dimensions.
✕The paper provides families of matrices over an arbitrary field k but does not give concrete examples for specific fields.

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