Quasi racks, quasi bijective and quasi non-degenerate set-theoretic solutions of the Yang-Baxter equation

This work initiates a systematic study of the class of quasi bijective and quasi non-degenerate solutions to the set-theoretic Yang-Baxter equation. The motivation stems from the observation that solutions that arise from dual weak braces belong to these classes. The notions of quasi rack and derived solution are introduced and examined, extending the classical definitions. Additionally, a family of quasi left non-degenerate solutions is described in terms of quasi racks and g-twists, analogous to the left non-degenerate case. Furthermore, we completely characterize a class of quasi racks that are Plonka sum of racks.

💡 Research Summary

This paper initiates a systematic study of two new classes of set‑theoretic solutions of the Yang–Baxter equation (YBE): quasi‑bijective and quasi‑non‑degenerate solutions. The motivation comes from the observation that solutions arising from dual weak braces naturally belong to these classes. After recalling the classical framework—set‑theoretic solutions r(x,y) = (λₓ(y), ρ_y(x)), the notions of left/right non‑degeneracy, and bijectivity—the authors introduce the concept of a quasi‑bijective solution. A solution (X,r) is quasi‑bijective if there exists a unique “relative inverse’’ r⁻ satisfying r r⁻ r = r, r⁻ r r⁻ = r⁻ and r r⁻ = r⁻ r. This weaker condition replaces the usual requirement r r⁻¹ = id and is shown to be satisfied by the solutions associated with any weak brace; in the case of a dual weak brace the associated maps λₓ and ρₓ form Clifford semigroups, guaranteeing the quasi‑bijective property.

The paper then defines quasi‑left‑non‑degenerate, quasi‑right‑non‑degenerate, and quasi‑non‑degenerate solutions. For a quasi‑left‑non‑degenerate solution each λₓ admits a unique relative inverse λₓ⁻ such that λₓ λₓ⁻ λₓ = λₓ, λₓ⁻ λₓ λₓ⁻ = λₓ⁻ and λₓ λₓ⁻ = λₓ⁻ λₓ. Moreover the “mixed’’ maps λ₀ₓ := λₓ λₓ⁻ = λₓ⁻ λₓ commute with all λ_y. Analogous conditions are imposed on the ρ‑maps for quasi‑right‑non‑degeneracy. These definitions generalize the classical non‑degenerate case (where λₓ and ρₓ are bijections) and capture the algebraic behaviour of dual weak braces.

To provide a structural counterpart, the authors introduce quasi‑racks. A quasi‑rack is a left shelf (X, ▷) whose left translations Lₓ are not necessarily bijective but satisfy the same semigroup‑inverse relations as the λ‑maps above: Lₓ Lₓ⁻ Lₓ = Lₓ, Lₓ⁻ Lₓ Lₓ⁻ = Lₓ⁻, Lₓ Lₓ⁻ = Lₓ⁻ Lₓ, and Lₓ Lₓ⁻ L_y = L_y Lₓ Lₓ⁻. If additionally Lₓ(x)=x for all x, the structure is called a quasi‑quandle. This definition extends the classical rack (where each Lₓ is a permutation) to the setting where the left translations form a Clifford semigroup inside the full transformation monoid.

A central contribution is the generalized derived solution associated with a quasi‑rack. For a shelf (X, ▷) define L₀ₓ := Lₓ Lₓ⁻. The derived map is

 r_▷(x, y) = ( L₀ₓ(y), L_y(x) ).

The authors identify three families of quasi‑racks that guarantee r_▷ is a quasi‑non‑degenerate solution:

1. Condition (∗) L₀ₓ L_y = L₀ₓ L₀_y for all x,y.
2. Condition (∗∗) L_y(x) = L_{L₀ₓ(y)}(x) for all x,y.
3. Condition (∗∗∗) L₀ₓ(x) = x for all x,y.

Each condition is independent; (∗∗∗) is the strongest and ensures that the derived solution is quasi‑bijective. The paper provides explicit examples for each family, and a computer‑generated enumeration (Table 1) of all quasi‑racks of size ≤ 4 up to isomorphism.

A striking structural result is that quasi‑racks satisfying both (∗) and (∗∗∗) can be realized as Plonka sums of racks. In other words, the underlying set decomposes into a semilattice of components, each of which carries a genuine rack structure; the global operation is obtained by “gluing’’ these racks along the semilattice. Consequently, the derived solution of such a quasi‑rack is precisely the strong semilattice (equivalently, Plonka sum) of the derived solutions of the constituent racks. This mirrors known results for racks but now holds in the broader quasi‑rack context.

Section 3 introduces g‑twists, a generalization of Drinfel’d twists. A map φ : X→Aut(X, ▷) is a twist if it satisfies the classical compatibility condition with the shelf operation. A g‑twist additionally requires that the left‑translation semigroup elements L₀ₓ commute with φ_y. Using a g‑twist φ and a family of maps gₓ, the authors construct a family of quasi‑left‑non‑degenerate solutions:

 r_{φ,g}(x, y) = ( gₓ(y), φ_{gₓ(y)}^{-1}( L_y(x) ) ).

When φ is a genuine Drinfel’d twist and gₓ = id, this reduces to the known left‑non‑degenerate construction; the extra flexibility allows the resulting r to be only quasi‑bijective and quasi‑non‑degenerate, matching the behaviour of solutions from dual weak braces.

The paper also contains several technical lemmas about the uniqueness of relative inverses in regular semigroups, and shows that the sets {λₓ} and {ρₓ} arising from a dual weak brace form Clifford subsemigroups of the full transformation monoid. For finite dual weak braces, the images of λₓ and ρₓ are bijective, reinforcing the quasi‑non‑degenerate nature of the associated solutions.

In summary, the authors achieve three major advances:

• Conceptual extension – By defining quasi‑bijective and quasi‑non‑degenerate solutions, they capture a broader class that includes, but is not limited to, solutions from dual weak braces.
• Algebraic framework – The introduction of quasi‑racks and their Plonka‑sum decomposition provides a structural language parallel to the classical rack/quandle theory, yet adapted to the semigroup‑inverse setting.
• Construction tools – Generalized derived solutions and g‑twists give systematic methods to build large families of quasi‑solutions, with explicit criteria (∗), (∗∗), (∗∗∗) that are easy to verify in concrete examples.

The work opens several avenues for future research: extending the theory to infinite sets, exploring categorical formulations, investigating connections with Hopf algebras and quantum groups, and applying the quasi‑framework to physical models where exact bijectivity is too restrictive. Overall, the paper significantly broadens the algebraic landscape surrounding set‑theoretic solutions of the Yang–Baxter equation.