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arxiv: 2603.26883 · v2 · submitted 2026-03-27 · 🧮 math.RA · cs.LO

Kleene and Stone algebras of rough sets induced by reflexive relations

Jouni J\"arvinen , S\'andor Radeleczki This is my paper

Pith reviewed 2026-05-14 22:13 UTC · model grok-4.3

classification 🧮 math.RA cs.LO
keywords rough setsKleene algebrasStone algebrasreflexive relationspseudocomplemented algebrasdouble Stone algebrascompletely distributive lattices
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The pith

The completion DM(RS) of rough sets from reflexive relations forms a regular pseudocomplemented Kleene algebra when spatial and completely distributive.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper studies Kleene and Stone algebra structures on the completion DM(RS) of rough sets induced by a reflexive relation on a set. It derives the conditions under which DM(RS) becomes a regular pseudocomplemented Kleene algebra and a completely distributive double Stone algebra, focusing on cases where the completion is a spatial and completely distributive lattice. The paper then characterizes the reflexive relations for which DM(RS) is a regular double Stone algebra, showing this occurs precisely when the structure matches that obtained from equivalence relations. These results extend earlier work on rough set algebras from equivalences, quasiorders, and tolerance relations to the wider class of reflexive relations.

Core claim

For a reflexive relation on a universe, the completion DM(RS) of the poset of rough sets is a regular pseudocomplemented Kleene algebra whenever DM(RS) is a spatial completely distributive lattice; it is a completely distributive double Stone algebra under the further conditions derived in the paper. DM(RS) forms a regular double Stone algebra exactly for those reflexive relations that produce the same algebraic structure as equivalence relations.

What carries the argument

The completion DM(RS) of the ordered set of rough sets induced by the reflexive relation, equipped with pseudocomplementation and Kleene negation when the lattice is spatial and completely distributive.

If this is right

  • DM(RS) carries a regular pseudocomplement and satisfies the Kleene identity whenever the lattice condition holds.
  • The algebra becomes a completely distributive double Stone algebra for reflexive relations meeting the additional derived conditions.
  • The regular double Stone algebra case arises exactly when the reflexive relation induces the same structure as an equivalence relation.
  • The results apply to all reflexive relations whose induced rough set poset completes to a spatial completely distributive lattice.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Reflexivity combined with the lattice completion appears sufficient to recover the algebraic properties previously known only for equivalences.
  • Similar completion techniques could be applied to other relational generalizations of rough sets to obtain the same algebra types.
  • Explicit construction of a non-transitive reflexive relation and direct verification of the DM(RS) operations would test the boundary of the conditions.

Load-bearing premise

The completion DM(RS) forms a spatial and completely distributive lattice for the reflexive relation under consideration.

What would settle it

A reflexive relation for which the completion DM(RS) is spatial and completely distributive yet fails to satisfy the axioms of a regular pseudocomplemented Kleene algebra.

Figures

Figures reproduced from arXiv: 2603.26883 by Jouni J\"arvinen, S\'andor Radeleczki.

Figure 1
Figure 1. Figure 1: RS is isomorphic to 2 × 3. paper, sets in figures are often denoted simply by the sequence of their elements (e.g., 123 for {1, 2, 3}). Our paper is structured as follows. In the next section, we present the basic algebraic and lattice-theoretic notions used in this work. In Section 3, the basic facts about rough approximation operators, the ordered set of rough sets, and its completion DM(RS) are re￾calle… view at source ↗
Figure 2
Figure 2. Figure 2: The Hasse diagrams of ℘(U) ▲ and RS = DM(RS). The ordered set RS is itself a lattice, so DM(RS) = RS. Its Hasse diagram can be found in [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The Hasse diagrams of ℘(U) ▲ and RS = DM(RS). According to (3.7), the set of atoms of RS is {({x} ▼ , {x} ▲ ) | {x} ▲ is an atom of ℘(U) ▲ }. Thus, RS also has only one atom which is ({1} ▼, {1} ▲) = (∅, {1}). This implies that for each element (∅, ∅) ̸= ρ ∈ RS, we have ρ ∗ = (∅, ∅) and ρ ∗∗ = (U, U). For these elements ρ, the Stone condition ρ ∗ ∨ ρ ∗∗ = (U, U) holds. Since (∅, ∅) ∗ = (U, U), (∅, ∅) also … view at source ↗
Figure 4
Figure 4. Figure 4: The Hasse diagram of RS. RS and is isomorphic to the product 2 × 2 × 3, where 2 and 3 are chains of two and three elements, respectively. Remark 7.8. Proposition 7.3 gives a simple method to generate regular double Stone algebras in terms of reflexive relations. We first define an irredundant covering C of U. Then, we attach to each element x ∈ U a set of B of C such that x ∈ B. Also, each set in C needs t… view at source ↗
read the original abstract

We consider Kleene and Stone algebras defined on the completion DM(RS) of the ordered set of rough sets induced by a reflexive relation. We focus on cases where the completion forms a spatial and completely distributive lattice. We derive the conditions under which DM(RS) is a regular pseudocomplemented Kleene algebra and a completely distributive double Stone algebra. Finally, we describe the reflexive relations for which DM(RS) forms a regular double Stone algebra, which is the same structure as in the case of equivalences. Our results generalise earlier findings on algebras of rough sets induced by equivalences, quasiorders, and tolerance relations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper considers Kleene and Stone algebras defined on the completion DM(RS) of the ordered set of rough sets induced by a reflexive relation. It focuses on cases where this completion forms a spatial and completely distributive lattice, derives conditions under which DM(RS) is a regular pseudocomplemented Kleene algebra and a completely distributive double Stone algebra, and characterizes the reflexive relations for which DM(RS) forms a regular double Stone algebra (the same structure as for equivalences). The results generalize earlier findings for equivalences, quasiorders, and tolerance relations.

Significance. If the derivations hold, this work meaningfully extends the algebraic theory of rough sets from equivalence relations to the broader class of reflexive relations under explicitly scoped lattice conditions. The generalization to regular double Stone algebras is a concrete strengthening of prior results and could support further applications in algebraic logic and approximation spaces.

minor comments (2)
  1. The abstract introduces DM(RS) without a brief parenthetical gloss on its meaning; adding one sentence would improve immediate readability for readers outside the subfield.
  2. The introduction would benefit from an explicit roadmap paragraph listing the main theorems (e.g., the conditions for the Kleene algebra and the characterization of the double Stone case) with forward references to their sections.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

Minor self-citation in generalization; core claims remain independent

full rationale

The derivation explicitly conditions all algebraic results on the external lattice property that DM(RS) is spatial and completely distributive; the conditions for Kleene and Stone structures are then derived directly from that assumption and the definition of rough-set completion. Generalization to reflexive relations is stated as holding exactly when the lattice property obtains, matching the case of equivalences without any reduction of new equations to prior fitted parameters or self-referential definitions. Self-citations to earlier work on equivalences and tolerances are present but not load-bearing for the central claims.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard domain assumptions in rough set theory and lattice theory, with no free parameters or invented entities introduced in the abstract.

axioms (2)
  • domain assumption The relation inducing the rough sets is reflexive.
    Explicitly stated in the title and abstract as the setting for the rough sets RS.
  • domain assumption The completion DM(RS) forms a spatial and completely distributive lattice.
    The paper focuses on cases where this holds, as a prerequisite for the algebraic structures.

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