[Paper Review] Transformation Digroups

Keqin Liu|ArXiv.org|Sep 16, 2004

Fuzzy and Soft Set Theory4 citations

TL;DR

This paper introduces the concept of a transformation digroup, demonstrating that every digroup is isomorphic to a subdigroup of a symmetric digroup defined on a Cartesian product of sets. By constructing a faithful representation using non-bijective transformations, the author establishes a Cayley-type theorem for digroups, proving that the algebraic structure of any digroup can be realized through transformation operations on a set.

ABSTRACT

We introduce the notion of a transformation digroup and prove that every digroup is isomorphic to a transformation digroup.

Motivation & Objective

• To define a transformation digroup as a structure of non-bijective transformations on a Cartesian product of two sets.
• To show that every digroup can be embedded into a symmetric digroup, generalizing Cayley's theorem to the context of digroups.
• To provide a representation theorem for digroups using transformation operations, ensuring structural isomorphism.
• To clarify the role of bar-units and one-sided inverses in the construction of transformation digroups.

Proposed method

• Define a transformation digroup on the Cartesian product Δ × Γ using left and right product operations derived from transformations.
• Construct a map λ from a digroup G to a symmetric digroup on Δ × Γ, using bar-units and left multiplication maps.
• Prove that the map λ is well-defined, injective, and preserves both the left and right product operations.
• Use the diassociative law and properties of bar-units to ensure consistency across transformations.
• Demonstrate that the image of λ forms a subdigroup isomorphic to the original digroup G.
• Leverage the uniqueness of left and right inverses with respect to bar-units to maintain algebraic integrity in the representation.

Experimental results

Research questions

• RQ1 Can every digroup be represented as a transformation digroup via a faithful homomorphism?
• RQ2 How can non-bijective transformations on a Cartesian product be structured to form a symmetric digroup?
• RQ3 What is the role of bar-units in defining the transformation representation of a digroup?
• RQ4 Is there a Cayley-type isomorphism theorem for digroups, analogous to the classical group case?
• RQ5 How do left and right products in a digroup correspond to transformation operations on a set?

Key findings

• Every digroup is isomorphic to a subdigroup of a symmetric digroup, establishing a representation theorem for digroups.
• The transformation digroup construction uses non-bijective maps on Δ × Γ, generalizing the concept of permutation groups.
• The map λ is injective and preserves both the left and right product operations, ensuring algebraic fidelity.
• The representation is independent of the choice of bar-unit, as shown by the invariance of left and right inverses under bar-unit change.
• When |Δ| = 1, the symmetric digroup reduces to the symmetric group on Γ, showing consistency with classical group theory.
• The construction provides a complete realization of digroup structure through transformation operations, fulfilling a Cayley-like embedding.