Invertibility in partially ordered nonassociative rings

Technical Summary: Invertibility in Partially Ordered Nonassociative Rings

Problem Statement
The paper addresses the problem of characterizing invertible elements within the specific algebraic and topological context of partially ordered nonassociative rings. While invertibility is a fundamental concept in ring theory—enabling division and the solution of equations—its behavior in structures lacking associativity and endowed with both order and topological properties requires rigorous investigation. The authors seek to determine the size and nature of the set of invertible elements, specifically focusing on two scenarios:

1. The invertibility of elements within the interval $]0, 1]$ in a partially ordered nonassociative ring.
2. The invertibility of elements $x$ in a nonassociative ring where a generalized seminorm $f$ satisfies $f(1-x) < 1$, within the framework of Hausdorff sequentially Cauchy-complete weak-quasi-topological rings.

Methodology
The authors employ a synthesis of order theory, nonassociative algebra, and topological group theory. The methodology relies on the following structural components:

• Algebraic Framework: The study is conducted on nonassociative rings (where multiplication is not necessarily associative) that are partially ordered. The rings are assumed to be unitary (possessing a multiplicative identity $1 \geq 0$).
• Topological Framework: The paper introduces and utilizes Frink's interval topology on partially ordered sets. This topology is defined by a subbase of closed sets consisting of rays $[a, +\infty[$ and $]-\infty, b]$. The authors establish that this topology renders certain ordered groups into locally convex quasi-topological groups.
• Generalized Seminorms: A novel concept is introduced: a seminorm $f$ mapping a nonassociative ring $R_1$ to a partially ordered nonassociative ring $R_2$ (endowed with Frink's interval topology). This map satisfies subadditivity, submultiplicativity, and evenness. The topology on $R_1$ is induced by this seminorm.
• Completeness and Convergence: The proofs rely on Hausdorff sequential Cauchy-completeness of the ring $R_1$ under the induced topology. The authors utilize the concept of monotone $\sigma$-completeness (existence of suprema for bounded increasing sequences) in the target ring $R_2$.
• Geometric Series Construction: The core mechanism for proving invertibility is the construction of "oriented geometric series." The authors define iterated powers $x^{\to n}$ (where $x^{\to 0}=1, x^{\to n+1} = x \cdot x^{\to n}$) and demonstrate that under specific conditions, the series $\sum_{n \in \mathbb{N}} x^{\to n}$ converges to an inverse.

Key Contributions and Results

1. Topological Properties of Seminormed Rings:

   • The paper establishes that a nonassociative unitary ring endowed with the topology induced by a norm into a totally ordered associative division ring (with Frink's interval topology) forms a Hausdorff locally convex quasi-topological group.
   • Furthermore, if the target ring is a division ring, the multiplication map in the source ring is shown to be continuous, leading to the definition of a weak-quasi-topological ring (a group that is quasi-topological with a separately continuous product).

2. Invertibility in Partially Ordered Nonassociative Rings (Theorem 3.1):

   • The authors prove that in a partially ordered nonassociative ring $R$ that is monotone $\sigma$-complete, if $1 \geq 0$ and every element in $]0, 1]$ admits a "right-sup-almost-inverse," then the interval $]0, 1]$ consists entirely of invertible elements.
   • The inverse is explicitly constructed as the limit of the series $\sum_{n \in \mathbb{N}} (1-x)^{\to n}$.
   • The paper notes that this result holds for standard rings like $\mathbb{R}$ and $\mathbb{Z}$, as well as polynomial rings with lexicographic ordering, but fails in rings lacking monotone $\sigma$-completeness or specific almost-inverse properties.

3. Invertibility in Seminormed Hausdorff Cauchy-Complete Rings (Theorem 3.2):

   • Extending the previous result, the authors show that if $f: R_1 \to R_2$ is a seminorm where $R_2$ is a partially ordered nonassociative ring with Frink's interval topology, and $R_1$ is Hausdorff sequentially Cauchy-complete under the topology induced by $f$, then any $x \in R_1$ satisfying $f(1-x) < 1$ is invertible.
   • The proof utilizes the convergence of the sequence of partial sums of $(1-x)^{\to n}$ in $R_1$, guaranteed by the completeness of $R_1$ and the properties of the series in $R_2$.

4. Original Topological Facts:

   • The paper demonstrates that Frink's interval topology on a totally ordered group is a topological group and that on a totally ordered division ring, it forms a topological ring.
   • It establishes the uniqueness of limits for increasing sequences in Frink's interval topology (Lemma 2.1).

Significance and Claims
The paper claims to contribute to the theory of invertibility by extending classical results (often based on metric spaces and associative algebras) to the broader, more complex setting of nonassociative rings equipped with partial orders and Frink's interval topology.

Specifically, the authors highlight:

• The introduction of a topology induced by a seminorm mapping into a partially ordered nonassociative ring, a concept they state has "never been introduced" to their knowledge.
• The proof that the interval $]0, 1]$ in suitable partially ordered nonassociative rings consists entirely of invertible elements.
• The generalization of the condition for invertibility ($f(1-x) < 1$) to rings that are not necessarily associative, provided they satisfy specific completeness and order-theoretic conditions.

The authors remain modest regarding the optimality of their results, noting that in the ring of integers $\mathbb{Z}$, only $1$ is invertible in $]0, +\infty[$, suggesting that the interval $]0, 1]$ is likely the maximal range for such a general theorem. They also clarify that their work addresses the existence and uniqueness of inverses within these specific structural contexts, rather than redefining the general notion of invertibility.