On Nondefinability of Interior-Connectedness via the Contact Relation

Technical Summary: On Nondefinability of Interior-Connectedness via the Contact Relation

Problem Statement
The paper addresses a fundamental question within the field of Boolean Contact Algebras (BCAs): the expressive power of the contact relation $C$ in capturing topological properties. While it is established that the contact relation can define standard connectedness ($c(x)$) within regular closed algebras, the authors investigate whether the stronger property of interior-connectedness (or internal connectedness)—where the interior of a region is a connected subspace—can be similarly defined. Previous work by Ivanova [9] demonstrated that interior-connectedness is not definable using the contact relation alone, employing Padoa's method to construct isomorphic contact algebras where one algebra contains an interior-connected element and the other does not. The specific problem this paper tackles is to analyze the minimality of such counterexamples. The authors seek to determine the minimal cardinalities of the underlying topological spaces and the regular closed algebras required to demonstrate this non-definability.

Methodology
The authors employ a combination of topological construction, algebraic isomorphism analysis, and combinatorial minimality proofs.

1. Graph-Based Heuristics: The authors utilize a five-point topological space (visualized as a graph with specific nodes and edges) to construct a counterexample. This space, denoted $T$, is designed with specific symmetries to facilitate the analysis of regular open and regular closed sets.
2. Padoa's Method: Following Ivanova's approach, the authors construct two isomorphic BCAs, $B_1$ and $B_2$, derived from topological spaces $X_1$ and $X_2$. They demonstrate that while the algebras are isomorphic via a map $f$, the property of interior-connectedness is not preserved under $f$.
3. Minimality Proofs: The core methodological contribution involves proving that the constructed examples are minimal. The authors systematically analyze the constraints on the number of elements in the algebras and the number of points in the underlying spaces. They utilize case analysis on the possible contact relations among atoms in four-point spaces to show that no smaller configuration can yield the required non-definability result.

Key Contributions and Results

• Simplified and Minimal Counterexample: The paper refines Ivanova's proof by presenting a construction based on a five-point space. The authors show that by removing a specific non-distinguished point ($a$) from the original space, they obtain a subspace with four points. The mapping between the regular closed algebra of the five-point space and the four-point subspace serves as a BCA-isomorphism that fails to preserve interior-connectedness. Specifically, a set $R$ in the five-point space has a connected interior, but its image $R \setminus \{a\}$ in the four-point space has a disconnected interior.

• Theorem on Algebra Cardinality (Theorem 3.1): The authors prove that for any BCA-isomorphism $f$ that fails to preserve interior-connectedness, both algebras involved must contain at least eight elements. They demonstrate that algebras with two or four elements cannot support the necessary structural differences to break interior-connectedness while maintaining isomorphism.

• Theorem on Space Cardinality (Theorem 4.1): The paper establishes that the minimal number of points required for the underlying topological spaces in such a counterexample is five for one space and four for the other. The authors rigorously rule out the possibility of constructing such a counterexample using only four-point spaces. They analyze all possible contact configurations on four-point sets (full contact, partial contact, and empty contact) and show that in all cases, the resulting algebras either preserve interior-connectedness or fail to provide the necessary distinction between the two isomorphic structures.

• Robustness of the Result: The authors note that their findings hold even when restricting the class of algebras to complete BCAs or those satisfying strong versions of the contact axiom (C4), as the counterexamples are finite.

Significance and Claims
The paper positions itself as a modest but precise contribution to the theory of Boolean Contact Algebras. It does not claim to solve the general problem of definability for all topological properties but specifically targets the minimality conditions for the non-definability of interior-connectedness.

The authors claim that their work clarifies the "phenomenon of nondefinability" by providing a graph-based setting that enhances topological intuition. By proving that the five-point/four-point construction is minimal in terms of both algebraic and spatial cardinality, the paper establishes a lower bound for the complexity required to demonstrate that interior-connectedness cannot be expressed via the contact relation. The authors explicitly state that they do not use graph topology in the formal sense but use graphs as heuristic devices with precise topological meanings. The work serves to solidify the understanding of the expressive limits of BCAs, confirming that while connectedness is definable, interior-connectedness requires a more complex language or additional predicates to be captured.