On Non-definability of Internal Connectedness via Contact Relation

TL;DR

The paper addresses whether internal connectedness can be defined purely in terms of the contact relation in Boolean Contact Algebras. It adopts a graph-inspired minimality approach and builds on Ivanova's prior result, using a concrete five-point counterexample to show that internal connectedness is not definable from contact alone, even under maximal contact. It further proves a minimality theorem: any pair of isomorphic BCAs with opposite internal connectedness properties must derive from spaces with at least five and four points, respectively, and that four-point configurations cannot realize the non-definability. The findings clarify the expressive limits of the contact language in BCAs and identify the smallest finite structures that witness the phenomenon.

Abstract

This short paper is a small contribution to the field of Boolean Contact Algebras. We analyze the non-definability of the property of internal connectedness, and we prove certain minimality conditions for algebras and spaces that can be used in demonstrating that the aforementioned property cannot be expressed by means of contact within regular closed algebras.

Figures (10)

• Figure 1: The five-point space.
• Figure 2: On the left, we have the frame of open subsets of $T$ with its regular elements marked in cyan; on the right, the Boolean algebra of regular closed subsets of the space.
• Figure 3: $R$ is an internally connected element of $\mathop{\mathrm{RC}}\nolimits(\tau)$.
• Figure 4: A four-point subspace $T\setminus\{a\}$ of the five-point space $T$.
• Figure 5: $\mathop{\mathrm{Int}}\nolimits_{\tau'}(R\setminus\{a\})$ is a discrete subspace of $T\setminus\{a\}$, so $R\setminus\{a\}$ cannot be internally connected.

Theorems & Definitions (6)

• Theorem 2.1
• Theorem 3.1
• proof
• Theorem 4.1
• Lemma 4.2
• proof