Completely Strongly Hollow Ideals

• Completely strongly hollow ideals are ideals in rings and lattices where any cover by a join forces containment in a single constituent ideal.
• They enable precise decompositions and minimal representations, mirroring prime or irreducible elements in module theory and lattice structures.
• Their study bridges algebra and analysis by characterizing thin sets and porosity using blow-up operations and decay conditions.

A completely strongly hollow ideal is a refinement of the classical hollow ideal concept, appearing in both commutative ring theory and multiplicative lattice theory. It encodes an extremal join-detecting property: rather than a set or element being merely excluded from “thick” regions, every attempt to cover it by a union (or join) already forces the ideal to be contained in one constituent. This property underlies strict decomposition phenomena in module and lattice theory, characterizes uniquely minimal representations, and delineates structural boundaries among porous and thin sets. The following sections synthesize recent foundational results, explicit characterizations, and implications for decompositions and applications.

1. Definitional Framework and Basic Properties

In a commutative ring $R$, the terminology for “strongly hollow” and “completely strongly hollow” ideals has precise algebraic and order-theoretic expressions.

• Strongly Hollow: An ideal $I$ is strongly hollow if $I \subseteq J + K$ implies $I \subseteq J$ or $I \subseteq K$ for any ideals $J, K$.
• Completely Strongly Hollow: The completely strong version strengthens this: $I \subseteq \sum_{\alpha \in A} J_\alpha$ entails $I \subseteq J_{\beta}$ for some $\beta \in A$.

Analogously, in a multiplicative lattice $Q$, an element $a$ is completely strongly hollow if whenever $a \leq \bigvee_{i \in I} x_i$, then $a \leq x_j$ for some $j \in I$ (Goswami et al., 21 Aug 2025). This isolates an element that cannot be “submerged” except by one “branch” of a join—signifying maximal join-irreducibility.

For porous sets, the ideal generated by completely strongly porous sets at $0$ (CSP) in $\mathbb{R}^+$ is a proper subideal of the intersection of maximal ideals generated by strongly porous sets (SP): $I(\text{CSP}) \subsetneq \hat{I}(\text{SP})$ (Bilet et al., 2014). This strict inclusion demarcates a finer notion of hollowness among thin sets.

2. Characterizations via Lattice-Theoretic Criteria

In C-lattices (where every element is a join of compact elements), several characterizations of completely strongly hollow elements are established.

• Join-Irreducibility: For element $a$ in $Q$, complete strong hollowness is equivalent to the property that, given any family $\{x_i\}$, $a \leq \bigvee_i x_i$ implies $a \leq x_j$ for some $j$.
• Invariant $\kappa(a)$: Defined as $\kappa(a) = \bigvee \{ x \in Q : a \not\leq x \}$, the criterion $a \not\leq \kappa(a)$ diagnoses complete strong hollowness for compact $a$ [(Goswami et al., 21 Aug 2025), Theorem 2.7].
• Quotient and Localization Stability: The property is preserved under quotient maps and localizations at maximal elements. Specifically, if $i \in Q$, then $a \vee i$ remains completely strongly hollow in $Q/i$.

In ideal lattices of commutative rings, minimal PS-hollow representations (see below) tightly correspond to completely strongly hollow ideals, which act as irreducible “atoms” in the decomposition theory.

3. Blow-Up Characterizations in Porosity Theory

A key analytic tool for studying completely strongly hollow ideals among porous sets is the $q$-blow up operation.

• For $E \subseteq \mathbb{R}^+$, define $E(q) = \bigcup_{x \in E} (q^{-1} x, q x)$ ($q > 1$).
• The chain $Cc_1(E(q)) = \{(a_i, b_i)\}$ consists of the first connected components ordered by increasing $a_i$.

Ideal Membership Criteria:

• $E \in \hat{I}(\text{SP})$ iff for all $q > 1$, $Cc_1(E(q))$ is infinite, and $\limsup_{i \to \infty} b_i/a_i < \infty$ holds [(Bilet et al., 2014), (6.8)].
• $E \in I(\text{CSP})$ iff, for every $q > 1$, the chain is infinite and two conditions are satisfied for some $q_0 > 1$ and $M \in \mathbb{N}$, for all $q > q_0$:

$$\limsup_{n \to \infty} b_n / a_n < \infty \quad \text{and} \quad \lim_{n \to \infty} \max\left\{ \frac{b_{n+j+1}}{a_{n+j}} : 0 \leq j \leq M \right\} = 0$$

[(7.14), (7.15)]. This enforced decay distinguishes the “completely hollow” CSP ideal from the broader SP ideal.

4. Module-Theoretic Perspective: PS-Hollow Representations

In module theory, the notion of PS-hollow submodules generalizes strong hollowness to submodule lattices.

• Definition: An $R$-submodule $N \subseteq M$ is PS-hollow if $N \subseteq I M + L$ implies $N \subseteq I M$ or $N \subseteq L$ for all ideals $I$ and submodules $L$ [(Abuhlail et al., 2018), (12)].
• Decomposition Results: Over Artinian rings, every PS-hollow representable module $M$ has a minimal PS-hollow representation (Proposition 2.10), and uniqueness is guaranteed under certain conditions (Theorems 2.17, 2.18).
• Implications: Minimal PS-hollow representation aligns with decomposition into completely strongly hollow “atoms”, mirroring the way that ideals factor in lattice theory.

Examples include second modules (all submodules are PS-hollow), the Prüfer group $\mathbb{Z}(p^\infty)$ (where every submodule is strongly hollow), and modules over $\mathbb{Z}_{pq}$ decomposed into minimal PS-hollow pieces.

5. Structural and Representational Consequences

Completely strongly hollow elements underpin decompositions in many multiplicative lattices and module categories.

• In Gelfand C-lattices, completely strongly hollow elements coincide with minimal or uniform elements [(Goswami et al., 21 Aug 2025), Theorem 3.4, Corollary 3.6].
• In weak Noether lattices, the property is equivalent to the lattice being a chain, allowing every element to factor as a finite product of completely strongly hollow elements [(Goswami et al., 21 Aug 2025), Propositions 5.3, 5.4].
• Representation: In many contexts, the unit element $1$ or an arbitrary lattice element admits a unique minimal join (or product) decomposition into completely strongly hollow elements, paralleling unique factorization in rings.

This suggests that completely strongly hollow ideals/elements serve as “building blocks” much like prime ideals or irreducible modules do in other algebraic settings.

6. Connections to Thin Set Theory and Analysis

The connection between hollowness and “thinness” is formalized by the paper of porosity and the behavior of cluster sets, quasiconformal mappings, Julia sets, and free boundary problems (Bilet et al., 2014). The precise geometric blow-up criteria for CSP ideals encode an analytic sparsity that is stronger than mere porosity—placing them at the core of problems involving lacunarity or fractal behavior near critical points.

A plausible implication is that similar hollow ideal structures may characterize sets of small Hausdorff dimension or fractal boundaries in geometric measure theory, providing structural insights into analytic contexts where “hollowness” imposes strong constraints.

7. Summary and Research Directions

Completely strongly hollow ideals unify several lines of research by refining join-detection and sparsity into an algebraic and lattice-theoretic principle. They are precisely characterized by join criteria in lattices, by blow-up decay in sets of real numbers, and by decomposition theorems in module categories. These constructs underlie uniqueness and minimality in representations and interact closely with localization, quotient operations, and structural features of both rings and lattices.

Current work suggests extensions to abstract ideal theory, lattice-theoretic generalizations, and analytic applications involving thin sets and porosity. The interplay between join-irreducibility, decomposition, and analytic sparsity continues to drive research on the role of completely strongly hollow ideals in both algebraic and geometric contexts.