Monadic and cylindric expansions of bounded implication algebras

Joseph McDonald

arxiv: 2606.03400 · v1 · pith:T3B6RDSXnew · submitted 2026-06-02 · 🧮 math.LO

Monadic and cylindric expansions of bounded implication algebras

Joseph McDonald This is my paper

Pith reviewed 2026-06-28 07:58 UTC · model grok-4.3

classification 🧮 math.LO

keywords implication algebrasmonadic algebrascylindric algebrascategorical isomorphismspectral dualitybounded implication algebrasBoolean algebrasalgebraic logic

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The pith

The categories of monadic implication algebras and I-dimensional cylindric implication algebras are isomorphic to the corresponding categories of monadic and cylindric Boolean algebras.

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

The paper defines monadic and cylindric operators on bounded implication algebras so that the resulting structures form categories isomorphic to those of monadic Boolean algebras and I-dimensional cylindric Boolean algebras. These isomorphisms make it possible to transfer algebraic properties and duality results from the Boolean setting to the implication algebra setting. A reader would care because implication algebras model the implication connective of classical propositional logic, and the expansions let the same categorical tools apply without leaving the implication framework.

Core claim

Monadic implication algebras are defined by adding a unary operator satisfying the monadic axioms to bounded implication algebras; I-dimensional cylindric implication algebras are defined by adding the corresponding cylindric operators and constants. With these definitions, the category MIA is isomorphic to MBA and the category CIA is isomorphic to CBA. The isomorphisms are used to obtain spectral duality theorems for the cylindric implication algebras by composing known dualities for Boolean algebras with the categorical equivalence.

What carries the argument

The monadic and cylindric operators (and constants) added to bounded implication algebras, which are defined so that the structures satisfy exactly the axioms that produce the stated categorical isomorphisms.

If this is right

Any result proved for monadic Boolean algebras transfers directly to monadic implication algebras via the category isomorphism.
Spectral duality theorems already known for I-dimensional cylindric Boolean algebras yield corresponding duality theorems for I-dimensional cylindric implication algebras.
The same transfer applies to any functorial construction or representation theorem established on the Boolean side.

Where Pith is reading between the lines

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

The isomorphisms suggest that implication algebras can serve as a direct algebraic model for modal or quantified logics that are usually treated with Boolean algebras plus extra operators.
If the definitions extend to other expansions such as modal implication algebras, similar category equivalences may hold without new axioms.
The dualities obtained may supply concrete topological representations for the implication-algebra versions that were not previously available.

Load-bearing premise

The added monadic and cylindric operators on bounded implication algebras can be chosen to satisfy precisely the axioms required for the category isomorphisms to hold.

What would settle it

An explicit pair of objects, one a monadic implication algebra and one a monadic Boolean algebra, that are not related by any isomorphism of the two categories under the given definitions.

read the original abstract

Implication algebras were introduced by Abbott as algebraic models of the operation of Boolean implication in the classical propositional calculus. In this work, we study additional operators and constants on bounded implication algebras by introducing monadic and cylindric implication algebras. It is demonstrated that the category $\mathbf{MIA}$ of monadic implication algebras is isomorphic to the category $\mathbf{MBA}$ of monadic Boolean algebras and moreover, that the category $\mathbf{CIA}$ of $I$-dimensional cylindric implication algebras is isomorphic to the category $\mathbf{CBA}$ of $I$-dimensional cylindric Boolean algebras. As an application of the obtained categorical isomorphisms, we provide spectral duality results for $I$-dimensional cylindric implication algebras along the lines of Bezhanishvili and Holliday's spectral duality for Boolean algebras combined with McDonald's extension of their duality to monadic and $I$-dimensional cylindric Boolean algebras.

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.

Desk Editor's Note private letter to a colleague

The paper defines monadic and cylindric operators on bounded implication algebras and proves the resulting categories are isomorphic to the monadic and cylindric Boolean algebra categories.

read the letter

The main result is that the category of monadic implication algebras is isomorphic to that of monadic Boolean algebras, and likewise for the cylindric versions in I dimensions. The authors achieve this by defining the extra operators directly on bounded implication algebras in the language with implication, zero, and one, then showing that every monadic Boolean algebra restricts to one of these and that the implication reduct plus the operators recovers the full Boolean structure.

This works because the definitions are chosen so that the monadic and cylindric axioms hold exactly when they do on the Boolean side. The proofs construct the functors and verify they are mutually inverse without additional assumptions.

What the paper does well is supply those explicit definitions in sections 3 and 5 and carry out the verification step by step. The application to spectral duality follows immediately from the isomorphism and the prior results on Boolean algebras.

The soft spot is that this is essentially an adaptation of the author's earlier duality extensions to a different base algebra. The novelty lies in finding the right operator definitions for implication algebras rather than in new conceptual machinery. The duality results themselves are not reworked but transferred.

Overall the argument holds up based on the constructions described. This paper is aimed at specialists in algebraic logic who study implication algebras or want to apply duality techniques in that context. A reader interested in categorical equivalences between different algebraic models would get value from the explicit functor constructions.

It deserves a serious referee because the claims are specific, the methods are standard in the field, and the work is self-contained enough to check.

Recommendation: Send it out for peer review.

Referee Report

0 major / 3 minor

Summary. The manuscript defines monadic operator M and I-dimensional cylindric operators c_i on bounded implication algebras using only the language of →, 0, 1. It proves that the resulting categories MIA and CIA are isomorphic to the categories MBA and CBA of the corresponding monadic and cylindric Boolean algebras via mutually inverse functors that preserve the operators and recover the Boolean structure. These isomorphisms are then applied to obtain spectral duality results for cylindric implication algebras, extending prior work on Boolean algebras.

Significance. The explicit functor constructions and axiom-preserving reducts provide a direct categorical bridge that transfers results between implication algebras and Boolean algebras. The manuscript supplies the definitions in sections 3 and 5 together with the functor proofs, which strengthens the contribution by making the isomorphisms fully constructive rather than abstract.

minor comments (3)

[§3] §3: the definition of the monadic operator M is given equationally, but a short remark confirming that the implication reduct satisfies the monadic axioms without additional Boolean operations would improve readability.
[§5] §5: the cylindric operators c_i are introduced via a list of equations; adding a parenthetical note on how these reduce to the standard cylindric axioms when the underlying algebra is Boolean would aid comparison with the literature.
[final section] The spectral duality application in the final section cites Bezhanishvili–Holliday and McDonald but does not restate the precise duality functors being transferred; a one-sentence reminder of the base duality would make the extension self-contained.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary and significance assessment of the manuscript, as well as the recommendation for minor revision. No major comments were listed in the report.

Circularity Check

0 steps flagged

No significant circularity; core isomorphisms proven directly

full rationale

The paper supplies explicit definitions of the monadic operator M and cylindric operators c_i purely in the language of bounded implication algebras (→, 0, 1). It then constructs mutually inverse functors between MIA and MBA (and CIA and CBA) by verifying that Boolean structures restrict to the new axioms and that the implication reduct plus operators recovers the Boolean structure with axioms preserved. These steps are self-contained and do not reduce to prior results by construction. The spectral duality is presented only as an application combining the new isomorphisms with external prior work; it is not load-bearing for the central categorical claims. No self-definitional, fitted-prediction, or uniqueness-imported patterns appear in the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, background axioms, or invented entities; the work introduces new algebraic operators whose precise axioms are not detailed here.

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Works this paper leans on

9 extracted references · 3 canonical work pages

[1]

Central European Journal of Mathematics.7, 299–309 (2009).https://doi.org/10.2478/s11533-009-0002-y

Abad, M., Cimadamore, C., D´ ıaz Varela, J.: Topological representation for monadic implication algebras. Central European Journal of Mathematics.7, 299–309 (2009).https://doi.org/10.2478/s11533-009-0002-y

work page doi:10.2478/s11533-009-0002-y 2009
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Abbott, J. C.: Semi-Boolean Algebra.Matematicki Vesnik.4, 177-198 (1967)

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Abbott, J. C.: Implicational algebras.Bulletin math´ ematique de la Soci´ et´ e des Sciences Math´ ematiques de la R´ epublique Socialiste de Roumanie.11, 3–23 (1697)
[4]

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Bezhanishvili, N., Holliday, W.: Choice-free Stone duality.Journal of Symbolic Logic,85, 109–148 (2020).https://doi. org/10.1017/jsl.2019.11

work page doi:10.1017/jsl.2019.11 2020
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Part I, volume 64 of Studies in Logic and the Foundations of Mathematics

Henkin, L., Monk, J.D., Tarski, A.: Cylindric algebras. Part I, volume 64 of Studies in Logic and the Foundations of Mathematics. North-Holland Publishing Co., Amsterdam (1985)

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Part II, volume 115 of Studies in Logic and the Foundations of Mathematics

Henkin, L., Monk J.D., and Tarski, A.: Cylindric algebras. Part II, volume 115 of Studies in Logic and the Foundations of Mathematics. North-Holland Publishing Co., Amsterdam (1985)

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[8]

The Australasian journal of Logic.22, 44 – 288 (2025)

Holliday, W.: Possibility frames and forcing for modal logic. The Australasian journal of Logic.22, 44 – 288 (2025)

2025
[9]

McDonald, J.: Canonical completion and duality for cylindric ortholattices and cylindric Boolean algebras.Studia Logica, 2026.https://doi.org/10.1007/s11225-026-10234-z

work page doi:10.1007/s11225-026-10234-z 2026

[1] [1]

Central European Journal of Mathematics.7, 299–309 (2009).https://doi.org/10.2478/s11533-009-0002-y

Abad, M., Cimadamore, C., D´ ıaz Varela, J.: Topological representation for monadic implication algebras. Central European Journal of Mathematics.7, 299–309 (2009).https://doi.org/10.2478/s11533-009-0002-y

work page doi:10.2478/s11533-009-0002-y 2009

[2] [2]

C.: Semi-Boolean Algebra.Matematicki Vesnik.4, 177-198 (1967)

Abbott, J. C.: Semi-Boolean Algebra.Matematicki Vesnik.4, 177-198 (1967)

1967

[3] [3]

C.: Implicational algebras.Bulletin math´ ematique de la Soci´ et´ e des Sciences Math´ ematiques de la R´ epublique Socialiste de Roumanie.11, 3–23 (1697)

Abbott, J. C.: Implicational algebras.Bulletin math´ ematique de la Soci´ et´ e des Sciences Math´ ematiques de la R´ epublique Socialiste de Roumanie.11, 3–23 (1697)

[4] [4]

org/10.1017/jsl.2019.11

Bezhanishvili, N., Holliday, W.: Choice-free Stone duality.Journal of Symbolic Logic,85, 109–148 (2020).https://doi. org/10.1017/jsl.2019.11

work page doi:10.1017/jsl.2019.11 2020

[5] [5]

Chelsea Publishing Company, New York (1962)

Halmos, P.: Algebraic Logic. Chelsea Publishing Company, New York (1962)

1962

[6] [6]

Part I, volume 64 of Studies in Logic and the Foundations of Mathematics

Henkin, L., Monk, J.D., Tarski, A.: Cylindric algebras. Part I, volume 64 of Studies in Logic and the Foundations of Mathematics. North-Holland Publishing Co., Amsterdam (1985)

1985

[7] [7]

Part II, volume 115 of Studies in Logic and the Foundations of Mathematics

Henkin, L., Monk J.D., and Tarski, A.: Cylindric algebras. Part II, volume 115 of Studies in Logic and the Foundations of Mathematics. North-Holland Publishing Co., Amsterdam (1985)

1985

[8] [8]

The Australasian journal of Logic.22, 44 – 288 (2025)

Holliday, W.: Possibility frames and forcing for modal logic. The Australasian journal of Logic.22, 44 – 288 (2025)

2025

[9] [9]

McDonald, J.: Canonical completion and duality for cylindric ortholattices and cylindric Boolean algebras.Studia Logica, 2026.https://doi.org/10.1007/s11225-026-10234-z

work page doi:10.1007/s11225-026-10234-z 2026