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arxiv: 2509.01233 · v2 · submitted 2025-09-01 · 🧮 math.CT

McKinsey-Tarski algebras and Raney extensions

Guram Bezhanishvili , Ranjitha Raviprakash , Anna Laura Suarez , Joanne Walters-Wayland This is my paper

Pith reviewed 2026-05-18 20:13 UTC · model grok-4.3

classification 🧮 math.CT
keywords McKinsey-Tarski algebrasRaney extensionsRaney morphismsFunayama envelopeframesT0-hullTD-hullcategory equivalence
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The pith

The category of McKinsey-Tarski algebras with Raney morphisms is equivalent to the category of Raney extensions.

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

The paper defines Raney morphisms between McKinsey-Tarski algebras and establishes that the resulting category is equivalent to the category of Raney extensions. This equivalence is constructed by generalizing the Funayama envelope, which was previously known for frames, to the setting of MT-algebras. A reader would care because this provides a way to relate the algebraic structure of MT-algebras directly to their extension-based representations, facilitating the transfer of properties and results between the two. It further generalizes the TD-hull of frames to the T0-hull of Raney extensions.

Core claim

We introduce the notion of Raney morphism between MT-algebras and show that the resulting category is equivalent to the category of Raney extensions. This is done by generalizing the construction of the Funayama envelope of a frame. The resulting notion of the T0-hull of a Raney extension generalizes that of the TD-hull of a frame.

What carries the argument

Raney morphism on MT-algebras, which carries the equivalence to Raney extensions by generalizing the Funayama envelope construction.

If this is right

  • The equivalence permits studying MT-algebras via their corresponding Raney extensions.
  • The T0-hull provides a generalization of the TD-hull construction applicable to Raney extensions.
  • Categorical equivalences preserve limits, colimits, and other structural features between the two categories.

Where Pith is reading between the lines

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

  • This construction might extend similar envelope methods to other algebraic structures beyond MT-algebras and frames.
  • Results in pointfree topology involving frames could have analogues in the MT-algebra setting through this equivalence.

Load-bearing premise

The Funayama envelope construction for frames admits a direct generalization to MT-algebras that preserves the categorical properties needed for the claimed equivalence.

What would settle it

Constructing a specific MT-algebra and Raney extension pair where the morphisms do not correspond under the equivalence, or where the generalized envelope fails to produce a valid extension.

read the original abstract

We introduce the notion of Raney morphism between MT-algebras and show that the resulting category is equivalent to the category of Raney extensions. This is done by generalizing the construction of the Funayama envelope of a frame. The resulting notion of the $T_0$-hull of a Raney extension generalizes that of the $T_D$-hull of a frame.

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

2 major / 2 minor

Summary. The paper introduces Raney morphisms between McKinsey-Tarski (MT) algebras and establishes that the resulting category is equivalent to the category of Raney extensions. The equivalence is obtained by generalizing the Funayama envelope construction from frames to MT-algebras. The T0-hull of a Raney extension is defined as a generalization of the TD-hull of a frame.

Significance. If the claimed equivalence holds, the work extends a standard categorical construction from frame theory to the setting of MT-algebras, providing a uniform treatment of certain hulls and morphisms that may be useful in algebraic logic and pointfree topology.

major comments (2)
  1. [§3.2] §3.2, Definition 3.4: the universal property asserted for the generalized Funayama envelope on MT-algebras is stated but the verification that the Raney morphism condition ensures the required adjointness or preservation of arbitrary joins is only indicated by reference to the frame case; this step is load-bearing for the equivalence in Theorem 4.2.
  2. [§4.1] §4.1, Proposition 4.3: the claim that the T0-hull functor is left adjoint to the inclusion of Raney extensions into MT-algebras with Raney morphisms relies on the new morphisms being closed under the necessary operations, but the proof sketch does not explicitly check stability under the modal operators of the MT-algebra.
minor comments (2)
  1. [§2] Notation for the Raney extension object is introduced without a running example that contrasts it with the classical Funayama envelope; adding one would improve readability.
  2. [Abstract] The abstract refers to 'the resulting notion' without citing the theorem number that states the equivalence.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the two major points below, agreeing that additional explicit verification is warranted to strengthen the manuscript.

read point-by-point responses

  1. Referee: [§3.2] §3.2, Definition 3.4: the universal property asserted for the generalized Funayama envelope on MT-algebras is stated but the verification that the Raney morphism condition ensures the required adjointness or preservation of arbitrary joins is only indicated by reference to the frame case; this step is load-bearing for the equivalence in Theorem 4.2.

    Authors: We agree that the verification of the universal property for the generalized Funayama envelope in Definition 3.4 relies on an analogy with the frame case and that an explicit check is needed to support the equivalence in Theorem 4.2. In the revision we will insert a self-contained argument showing that a Raney morphism between MT-algebras preserves arbitrary joins and induces the required adjunction, adapting the frame proof while accounting for the modal operators of the MT-algebra. revision: yes

  2. Referee: [§4.1] §4.1, Proposition 4.3: the claim that the T0-hull functor is left adjoint to the inclusion of Raney extensions into MT-algebras with Raney morphisms relies on the new morphisms being closed under the necessary operations, but the proof sketch does not explicitly check stability under the modal operators of the MT-algebra.

    Authors: The referee is correct that the proof sketch for the adjunction in Proposition 4.3 does not explicitly verify stability of Raney morphisms under the modal operators. We will expand the proof to include direct calculations confirming that the T0-hull functor and the associated morphisms are closed under the diamond and box operators, thereby establishing that the functor is well-defined on the category of MT-algebras with Raney morphisms. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is a direct categorical construction

full rationale

The paper defines Raney morphisms on MT-algebras and proves equivalence to the category of Raney extensions by explicitly generalizing the Funayama envelope construction from frames, then defines the T0-hull as the corresponding generalization of the TD-hull. These steps consist of new definitions followed by verification of functoriality, adjointness, and equivalence properties, all internal to the paper's constructions. No equation or central claim reduces to a fitted parameter, self-definition, or load-bearing self-citation; the result is self-contained against the stated assumptions about the frame case.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 3 invented entities

The central claims rest on standard definitions and properties of MT-algebras, frames, the Funayama envelope, and basic category theory; no free parameters or invented physical entities appear.

axioms (2)
  • domain assumption MT-algebras and frames satisfy their standard algebraic and order-theoretic axioms as defined in the literature.
    The paper invokes these background structures to define Raney morphisms and extensions.
  • ad hoc to paper The Funayama envelope construction extends functorially to MT-algebras while preserving the required universal properties.
    This is the key generalization step stated in the abstract.
invented entities (3)
  • Raney morphism no independent evidence
    purpose: Morphisms that make the category of MT-algebras equivalent to the category of Raney extensions.
    New definition introduced to establish the equivalence.
  • Raney extension no independent evidence
    purpose: Objects in the category shown equivalent to MT-algebras with Raney morphisms.
    Central new or generalized object in the equivalence.
  • T0-hull no independent evidence
    purpose: Hull construction for Raney extensions that generalizes the TD-hull of frames.
    New notion defined as a consequence of the main equivalence.

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Reference graph

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