[Paper Review] Normed \Omega-Group

Aleks Kleyn|arXiv (Cornell University)|May 15, 2013

Aerospace Engineering and Control Systems5 references3 citations

TL;DR

This paper introduces the concept of a normed Ω-group, a universal algebraic structure generalizing groups and rings by endowing an Ω-algebra with a norm that induces a topology and enables continuity analysis. It establishes foundational topological and algebraic properties, including convergence, completeness, and continuous representations, and extends the framework to Ω-rings and modules, demonstrating that normed Ω-groups support a robust calculus-like structure through continuous operations and representations.

ABSTRACT

Since sum which is not necessarily commutative is defined in \\Omega-algebra A, then \\Omega-algebra A is called \\Omega-group. I also considered representation of \\Omega-group. Norm defined in \\Omega-group allows us to consider continuity of operations and continuity of representation.

Motivation & Objective

• To formalize the concept of a normed Ω-group as a generalization of groups and rings with a norm-induced topology.
• To investigate the topological properties of Ω-groups, including convergence, completeness, and continuity of operations.
• To develop a theory of continuous representations of Ω-groups, particularly in complete Ω-groups.
• To extend the framework to Ω-rings and modules by introducing product operations and effective representations.
• To establish conditions under which representations and operations remain continuous under limits and completions.

Proposed method

• Defines an Ω-group as an algebraic structure with a non-commutative sum and polyadditive operations, generalizing groups and rings.
• Introduces a norm on an Ω-group satisfying standard axioms (non-negativity, definiteness, triangle inequality, absolute homogeneity), turning it into a normed Ω-group.
• Uses the norm to define open and closed balls, and to formalize convergence of sequences via the ε–N criterion.
• Defines the norm of an n-ary operation as the supremum of the ratio of the norm of the output to the product of input norms.
• Applies the norm to analyze continuity of operations and representations, using limits and completeness to extend structures.
• Establishes that representations of complete Ω-groups into complete Ω-groups are continuous if they preserve limits and operations.

Experimental results

Research questions

• RQ1 How can a norm be consistently defined on a non-abelian, non-commutative universal algebra to induce a topology?
• RQ2 What conditions ensure that operations in an Ω-group are continuous under the norm-induced topology?
• RQ3 Can representations of Ω-groups be made continuous, and under what conditions do they preserve limits?
• RQ4 How does the completion of a normed Ω-group preserve its algebraic and topological structure?
• RQ5 What algebraic structures emerge when a product operation is introduced in a normed Ω-group, and how do they relate to rings and modules?

Key findings

• The norm on an Ω-group satisfies the reverse triangle inequality: ||a − b|| ≥ |||a| − |b|||, ensuring metric-like behavior.
• The norm of an n-ary operation ω is defined as ||ω|| = sup ||a₁…aₙω|| / (||a₁||…||aₙ||), and it bounds the output norm via ||a₁…aₙω|| ≤ ||ω|| ||a₁||…||aₙ||.
• Convergence in a normed Ω-group is defined via the standard ε–N criterion, and limits are unique under the norm topology.
• The completion of a normed Ω-group results in a complete normed Ω-group, preserving the algebraic and topological structure.
• A continuous representation of a complete Ω-group into another complete Ω-group is characterized by the preservation of limits and operation norms.
• When a product is introduced via a representation, the resulting structure satisfies distributivity and becomes an Ω-ring, with effective representations defining modules.