On Hom-Analogues of Heaps and Trusses

Technical Summary: Hom-Analogues of Heaps and Trusses

Problem Statement
The paper addresses the need to extend the algebraic theories of heaps, trusses, and braces into the Hom-algebraic setting. While classical heaps (ternary structures generalizing groups), trusses (heaps equipped with a distributive multiplication), and braces (structures linking group and ring theory via distributive laws) are well-established, their counterparts under "twisted" identities—where axioms hold only up to a distinguished self-map $\alpha$—remain underdeveloped. Specifically, the authors aim to define Hom-heaps, Hom-trusses, and Hom-braces, establish their structural properties, and clarify the correspondences between these twisted structures and their classical and Hom-group counterparts.

Methodology
The authors employ a constructive and axiomatic approach rooted in Hom-algebra theory. The methodology involves:

1. Definition and Axiomatization: Introducing formal definitions for Hom-heaps, Hom-trusses (in three distinct types), and Hom-braces (in three distinct types). These definitions adapt classical axioms (associativity, Mal'cev identities, distributivity) by incorporating a bijective twisting map $\alpha$ (and potentially a second map $\beta$).
2. Retraction and Construction: Investigating the relationship between Hom-heaps and Hom-groups by defining "retracts" (constructing a binary operation from a ternary one at a specific point). The authors analyze the conditions under which a retract yields a Hom-group, specifically requiring the base point to be a fixed point of the twisting map.
3. Equivalence Proofs: Establishing equivalences between the three proposed types of Hom-trusses and Hom-braces. This involves proving that under specific conditions (e.g., $\alpha = \beta$ or $\alpha = \text{id}$), the definitions collapse into one another or reduce to known structures.
4. Correspondence Analysis: Demonstrating how Hom-trusses naturally induce Hom-braces (and vice versa) when specific structural conditions (such as the multiplicative part forming a Hom-group or the existence of an absorber) are met.
5. Translation Maps: Utilizing translation maps in Hom-heaps to prove isomorphisms between Hom-group retracts formed at different fixed base points.

Key Contributions and Results

• Hom-Heaps and Hom-Groups:

  • The paper defines a Hom-heap as a set with a ternary operation and a bijective map $\alpha$ satisfying Hom-Mal'cev and Hom-associativity properties.
  • Theorem: The retract of a Hom-heap at a point $e$ forms a Hom-group if and only if $e$ is a fixed point of $\alpha$ (i.e., $\alpha(e) = e$).
  • The authors prove that translation maps between different fixed base points induce isomorphisms between the resulting Hom-groups, extending the classical result for heaps.
  • A bijective correspondence is established: every Hom-group induces a Hom-heap, and every Hom-heap with a fixed point induces a Hom-group.

• Hom-Trusses:

  • Three types of Hom-trusses are introduced:
    • Type (0): Based on a single twisting map $\alpha$ acting on both the heap and multiplication structures.
    • Type (1): Involves two commuting maps $\alpha$ and $\beta$, where $\alpha$ twists the heap and $\beta$ twists the multiplication, with specific distributive laws.
    • Type (2): A more complex variant involving higher-order compositions of $\alpha$ and $\beta$ in the associativity and distributivity laws.
  • Equivalence: It is shown that an $\alpha$-Hom-truss of Type (1) is equivalent to an $\alpha$-Hom-truss of Type (2) when $\alpha = \beta$. Furthermore, Type (0) is recovered when $\alpha = \text{id}$ in Type (1).
  • Construction: The paper demonstrates how to construct Hom-trusses from classical trusses by twisting operations with commuting automorphisms, and conversely, how to recover classical trusses from Hom-trusses via inverse twisting.

• Hom-Braces:

  • Analogous to trusses, three types of Hom-braces are defined, linking two group structures (one abelian, one Hom-group) via distributive laws.
  • Correspondence with Trusses: The authors prove that every Hom-truss of Type (0) where the multiplicative part is a Hom-group gives rise to a Hom-brace of Type (0). Conversely, a unital Hom-brace yields a unital Hom-truss.
  • Similar correspondences are established for Type (1) and Type (2) structures, particularly involving idempotent conditions and the existence of absorbers (elements that annihilate multiplication).

• Substructures: The paper defines Hom-subheaps and normal Hom-subheaps, providing criteria for normality and establishing properties related to quotient structures.

Significance
The paper claims to provide a unified framework that extends heap and truss theory to the Hom-algebraic setting. By systematically defining these structures and proving their interrelations, the work generalizes classical classification and extension results. The authors position this research as a foundational step toward the study of "Hom-affgebras" (Hom-affine algebras), which serve as the affine counterparts to Hom-algebras.

The significance lies in:

1. Generalization: Successfully adapting the rich machinery of heaps, trusses, and braces to the twisted context of Hom-algebras.
2. Unification: Showing that the three types of Hom-trusses and Hom-braces are interconnected, reducing to classical or simpler Hom-structures under specific constraints.
3. Potential Applications: The authors note that these structures have potential applications in the study of set-theoretic solutions to the Yang–Baxter equation and non-associative geometry, continuing the trajectory established by Brzeziński and Rump in the classical setting. The paper does not claim to solve specific Yang–Baxter problems but rather to lay the algebraic groundwork for such investigations.