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arxiv: 2605.21667 · v1 · pith:43SCDN5Znew · submitted 2026-05-20 · 🧮 math.LO · math.CT

From Multirelations to Meet-Relations: A Relational Duality for Semilattices with Adjunctions

William Zuluaga , Bel\'en Gimenez This is my paper

Pith reviewed 2026-05-22 08:15 UTC · model grok-4.3

classification 🧮 math.LO math.CT
keywords semilattices with adjunctionsmeet-relationsrelational dualitymultirelationsnormal mS-spacescategory isomorphismdual equivalence
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The pith

Semilattices with adjunctions admit a dual equivalence to spaces defined by meet-relations

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

The paper develops a relational duality for semilattices with adjunctions based on binary meet-relations. It establishes a dual equivalence between the category SLata and the category RelSP of spaces with A-relations. By defining normal mS-spaces, it demonstrates that this meet-relation approach canonically recovers the multirelational semantics, proving the two categories of spaces are isomorphic.

Core claim

We prove a dual equivalence between SLata and RelSP. Under the normality condition on mS-spaces, the multirelational structure can be canonically recovered from a meet-relation and conversely. As a consequence, the categories RelSP and SLataSp are isomorphic.

What carries the argument

A-relations, which are binary relations compatible with the semilattice structure and adjunctions, that define the relational semantics for SLatas.

If this is right

  • SLata is dually equivalent to RelSP.
  • RelSP and SLataSp are isomorphic categories.
  • Normal mS-spaces permit canonical translation between multirelations and meet-relations.
  • Modal semilattices are dually equivalent to MoS-spaces.

Where Pith is reading between the lines

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

  • This framework may allow simpler relational proofs for properties of adjunctions in semilattices.
  • The normality condition could be a key property distinguishing general and canonical cases in related dualities.

Load-bearing premise

The normality condition on mS-spaces is sufficient for the canonical recovery of the multirelational structure from a meet-relation and conversely.

What would settle it

A specific non-normal mS-space in which the meet-relation fails to uniquely determine the original multirelation would falsify the recovery result.

read the original abstract

We develop a relational duality for semilattices with adjunctions (SLatas) based on binary meet-relations. First, we introduce the category of MoS-spaces and establish a dual equivalence with modal semilattices. Then, by means of A-relations, we define the category RelSP and prove a dual equivalence between SLata and RelSP. To compare this framework with the multirelational semantics previously developed for SLatas, we introduce the notion of normal mS-space and show that, under this condition, the multirelational structure can be canonically recovered from a meet-relation, and conversely. As a consequence, we prove that the categories RelSP and SLataSp are isomorphic.

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 / 3 minor

Summary. The paper develops a relational duality for semilattices with adjunctions (SLatas) based on binary meet-relations. It introduces the category of MoS-spaces and establishes a dual equivalence with modal semilattices. Using A-relations, it defines the category RelSP and proves a dual equivalence between SLata and RelSP. It then introduces normal mS-spaces to show that multirelational structures can be canonically recovered from meet-relations (and conversely), yielding an isomorphism between RelSP and SLataSp.

Significance. If the dual equivalences hold, the manuscript supplies a useful bridge between multirelational and meet-relational semantics for SLatas. The explicit functors in both directions, together with the verification that they are inverse on objects and morphisms once normality is imposed, constitute a clear strength and allow direct canonical recovery of the two relational structures.

minor comments (3)
  1. The introduction would benefit from a short comparison table contrasting the new meet-relation framework with the earlier multirelational semantics for SLatas.
  2. Notation for A-relations and the induced meet-relations is introduced in §3 but could be cross-referenced more explicitly when the functors are defined in §4.
  3. A brief remark on whether the normality condition can be weakened while still guaranteeing the isomorphism would clarify the scope of the final result.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive evaluation of the manuscript. The referee summary accurately reflects the main results: the dual equivalence between SLata and RelSP via meet-relations and A-relations, together with the isomorphism between RelSP and SLataSp once normal mS-spaces are imposed. We appreciate the recommendation of minor revision and stand ready to implement any editorial or presentational adjustments.

Circularity Check

0 steps flagged

No significant circularity; dual equivalences proven via explicit functors and direct verification

full rationale

The manuscript defines new categories (MoS-spaces, RelSP, normal mS-spaces) and constructs explicit functors establishing dual equivalences SLata ≃ RelSP and the isomorphism RelSP ≅ SLataSp under the normality condition. It verifies that the functors are inverses on objects and morphisms and that meet-relations and multirelations recover each other canonically. These steps rely on direct relational compositions and adjunction preservation rather than any self-definitional reduction, fitted parameter renamed as prediction, or load-bearing self-citation chain. The comparison to prior multirelational semantics is presented as an independent consequence once normality is imposed, leaving the core derivations self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract does not introduce or rely on any free parameters, ad-hoc axioms, or invented entities beyond standard background notions in category theory and order theory.

pith-pipeline@v0.9.0 · 5657 in / 1066 out tokens · 50784 ms · 2026-05-22T08:15:47.708464+00:00 · methodology

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Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages

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    Calomino, P

    I. Calomino, P. Mench´ on and W. J. Z. Botero. A topological duality for monotone expansions of semilattices. Applied Categorical Structures, 30, 1257–1282, 2022. https://doi.org/10.1007/ s10485- 022-09690-0

    work page 2022

  2. [2]

    Davey, H

    B. Davey, H. Priestley, H. A. Introduction to lattices and order. Cambridge university press (2002)

    work page 2002

  3. [3]

    Gim´ enez, G

    B. Gim´ enez, G. Pelaitay, W. Zuluaga. A Stone-type duality for semilattices with adjunctions. Journal of Logic and Computation, Volume 35, Issue 3, (2025)

    work page 2025

  4. [4]

    Appl Categor Struct 28, 853–875 (2020)

    Celani, S.A., Gonz´ alez, L.J.A Categorical Duality for Semilattices and Lattices. Appl Categor Struct 28, 853–875 (2020)

    work page 2020

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    Dunn, J.M., Gehrke, M., Palmigiano, A.: Canonical extensions and relational completeness of some substructural logics. J. Symbolic Logic. 70 (3), 713–740 (2005)

    work page 2005

  6. [6]

    Order 31, 321—341 (2014)

    Celani, S.A., Mench´ on, M.P.Monotonic Distributive Semilattices. Order 31, 321—341 (2014). ———————————————————————————– Bel´ en Gimenez, Departamento de Matem´ atica, Facultad de Ciencias Exactas (UNLP), 50 y 115, La Plata (1900) belengim.28@gmail.com ———————————————————————————– William Zuluaga, Facultad de Ciencias Exactas (UNCPBA), Pinto 399, Tandil (...

    work page 2014