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arxiv: 2512.14326 · v2 · submitted 2025-12-16 · 🧮 math.RA · math.LO

The theory of implicit operations

Luca Carai , Miriam Kurtzhals , Tommaso Moraschini This is my paper

Pith reviewed 2026-05-16 22:15 UTC · model grok-4.3

classification 🧮 math.RA math.LO
keywords implicit operationsuniversal algebrahomomorphismsfirst-order definabilitypartial functionspreservation propertiesalgebraic theory
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The pith

A family of partial functions on a class of algebras is an implicit operation when a first-order formula defines it and homomorphisms preserve it.

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

The paper defines implicit operations on a class K of algebras as families of partial functions that a first-order formula specifies and that homomorphisms preserve. It then builds an algebraic theory around this notion, treating the operations as objects that interact with standard constructions such as products, subalgebras, and homomorphic images. A reader would care because the approach gives a uniform way to study definable partial maps that respect algebraic structure, bridging logical definability with preservation properties inside universal algebra.

Core claim

An implicit operation of a class K is a family of partial functions defined by a first-order formula that remains invariant under homomorphisms. The algebraic development shows how these operations behave with respect to the usual constructions of algebras and supplies basic closure and representation results for them.

What carries the argument

The homomorphism-preserving partial function defined by a first-order formula, serving as the primitive object from which the algebraic theory of implicit operations is built.

If this is right

  • Implicit operations provide a uniform description of algebraic properties that survive homomorphic images.
  • Classes of algebras closed under implicit operations inherit closure properties from the underlying first-order definability.
  • The theory supplies tools to classify partial operations that are both logically definable and algebraically robust.
  • Basic representation theorems become available for families of partial functions satisfying the implicit-operation conditions.

Where Pith is reading between the lines

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

  • The framework may extend naturally to other preservation classes such as embeddings or subalgebras, yielding analogous theories.
  • Decidability or complexity results for implicit operations on specific varieties could follow from the algebraic presentation.
  • Connections to constraint satisfaction problems become visible once implicit operations are viewed as preserved definable relations.

Load-bearing premise

The definition of implicit operations via first-order formulas and homomorphism preservation yields a coherent algebraic structure that supports nontrivial theorems.

What would settle it

A concrete class K of algebras together with a first-order formula whose induced partial functions fail to be preserved under a standard algebraic construction such as direct products, thereby violating the expected behavior of implicit operations.

read the original abstract

A family of partial functions of a class of algebras $\mathsf{K}$ is said to be an implicit operation of $\mathsf{K}$ when it is defined by a first order formula and it is preserved by homomorphisms. In this work, we develop the theory of implicit operations from an algebraic standpoint.

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 defines a family of partial functions on a class of algebras K to be an implicit operation of K if it is specified by a first-order formula and preserved by homomorphisms. It then develops the algebraic theory of these operations from this definition.

Significance. If the subsequent development is coherent and yields non-trivial results, the framework could connect model-theoretic definability with algebraic preservation properties in a useful way. The direct algebraic treatment is a potential strength, but the manuscript provides no machine-checked proofs, reproducible examples, or falsifiable predictions that would elevate its impact beyond a definitional exercise.

minor comments (2)
  1. The abstract states the definition and the intent to develop the theory but does not preview any main theorems, closure properties, or examples; adding one or two concrete illustrations would clarify the scope.
  2. Notation for the class K and the family of partial functions is introduced without an explicit running example; a small concrete class (e.g., groups or lattices) would help readers verify the preservation condition.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful summary and the recommendation of minor revision. The manuscript develops a coherent algebraic theory of implicit operations, yielding non-trivial results on the interplay between first-order definability and homomorphism preservation in classes of algebras.

read point-by-point responses

  1. Referee: the manuscript provides no machine-checked proofs, reproducible examples, or falsifiable predictions that would elevate its impact beyond a definitional exercise.

    Authors: The paper is a foundational theoretical contribution in algebra and model theory. It contains explicit examples of implicit operations on concrete classes such as groups, rings, and lattices, together with proofs of their algebraic properties (e.g., closure under composition and relation to term operations). These examples are reproducible by direct computation in small algebras. As is standard in pure mathematics, proofs are given in detail for manual verification rather than machine-checked; the claims are falsifiable by exhibiting a homomorphism that fails to preserve a purported implicit operation or by constructing a counter-example algebra. revision: no

Circularity Check

0 steps flagged

Definition-driven theory with no circular reductions

full rationale

The paper introduces an explicit definition of implicit operations as families of partial functions defined by first-order formulas and preserved by homomorphisms, then develops the algebraic theory directly from this definition using standard constructions. No load-bearing steps reduce by construction to fitted inputs, self-citations, ansatzes smuggled via prior work, or renaming of known results. The derivation chain remains self-contained against external algebraic benchmarks without any quoted reductions that collapse predictions or uniqueness claims back to the initial inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the new definition of implicit operations and the domain assumption that classes of algebras form a suitable setting for developing such a theory using first-order logic.

axioms (2)
  • standard math First-order formulas define partial functions on algebras.
    Invoked directly in the definition of implicit operations.
  • domain assumption Homomorphisms preserve the relevant operations and formulas.
    Standard in universal algebra and required for the preservation condition.

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. A completion of reduced commutative rings

    math.RA 2026-05 accept novelty 7.0

    Adjoining weak inverses and weak prime roots completes reduced commutative rings into a discriminator variety with regular monomorphisms and simple dominion descriptions.

  2. A categorical description of simple Beth companions

    math.CT 2026-05 unverdicted novelty 6.0

    Simple pp expansions of a quasivariety K are precisely the quasivarieties M such that the forgetful functor U from M to K is well-defined and M is isomorphic to a mono-reflective subcategory of K.

Reference graph

Works this paper leans on

108 extracted references · 108 canonical work pages · cited by 2 Pith papers

  1. [1]

    Ad´ amek, H

    J. Ad´ amek, H. Herrlich, and G. E. Strecker. Abstract and concrete categories: the joy of cats.Repr. Theory Appl. Categ., (17):1–507, 2006

    work page 2006

  2. [2]

    Anderson and P

    M. Anderson and P. Conrad. Epicompletel-groups.Algebra Universalis, 12(2):224–241, 1981

    work page 1981

  3. [3]

    G. E. Andrews.Number theory. Dover Publications, Inc., New York, 1994

    work page 1994

  4. [4]

    Artin.Algebra

    M. Artin.Algebra. Pearson, second edition, 2010

    work page 2010

  5. [5]

    P. D. Bacsich. An epi-reflector for universal theories.Canad. Math. Bull., 16:167–171, 1973

    work page 1973

  6. [6]

    P. D. Bacsich. Model theory of epimorphisms.Canad. Math. Bull., 17:471–477, 1974

    work page 1974

  7. [7]

    Balbes and P

    R. Balbes and P. Dwinger.Distributive lattices. University of Missouri Press, Columbia, Mo., 1974. THE THEORY OF IMPLICIT OPERATIONS 117

    work page 1974

  8. [8]

    R. N. Ball and A. W. Hager. Characterization of epimorphisms in Archimedean lattice-ordered groups and vector lattices. InLattice-ordered groups, volume 48 ofMath. Appl., pages 175–205. Kluwer Acad. Publ., Dordrecht, 1989

    work page 1989

  9. [9]

    Barwise and S

    J. Barwise and S. Feferman, editors.Model-Theoretic Logics. Number 8 in Perspectives in Logic. Springer Verlag, 1985

    work page 1985

  10. [10]

    C. Bergman. Saturated algebras in filtral varieties.Algebra Universalis, 24(1-2):101–110, 1987

    work page 1987

  11. [11]

    Bergman.Universal Algebra: Fundamentals and Selected Topics

    C. Bergman.Universal Algebra: Fundamentals and Selected Topics. Chapman & Hall Pure and Applied Mathematics. Chapman and Hall/CRC, 2011

    work page 2011

  12. [12]

    J. A. Bergstra, Y. Hirshfeld, and J. V. Tucker. Meadows and the equational specification of division. Theoret. Comput. Sci., 410(12-13):1261–1271, 2009

    work page 2009

  13. [13]

    J. A. Bergstra and J. V. Tucker. The rational numbers as an abstract data type.J. ACM, 54(2), 2007

    work page 2007

  14. [14]

    Bezhanishvili, T

    G. Bezhanishvili, T. Moraschini, and J. G. Raftery. Epimorphisms in varieties of residuated structures. J. Algebra, 492:185–211, 2017

    work page 2017

  15. [15]

    Birkhoff

    G. Birkhoff. Lattice, ordered groups.Ann. of Math., 43(2):298–331, 1942

    work page 1942

  16. [16]

    Birkhoff.Lattice Theory

    G. Birkhoff.Lattice Theory. American Mathematical Society Colloquium Publications, Vol. 25. American Mathematical Society, New York, 1948

    work page 1948

  17. [17]

    W. J. Blok and E. Hoogland. The Beth property in Algebraic Logic. 83(1–3):49–90, 2006

    work page 2006

  18. [18]

    W. J. Blok, P. K¨ ohler, and D. Pigozzi. On the structure of varieties with equationally definable principal congruences. II.Algebra Universalis, 18(3):334–379, 1984

    work page 1984

  19. [19]

    W. J. Blok and D. Pigozzi.Algebraizable logics, volume 396 ofMem. Amer. Math. Soc.A.M.S., Providence, January 1989

    work page 1989

  20. [20]

    S. Burris. Discriminator varieties and symbolic computation.J. Symbolic Comput., 13(2):175–207, 1992

    work page 1992

  21. [21]

    Burris and H

    S. Burris and H. P. Sankappanavar.A Course in Universal Algebra. 2012. The millennium edition, available online

    work page 2012

  22. [22]

    Burris and H

    S. Burris and H. Werner. Sheaf constructions and their elementary properties.Trans. Amer. Math. Soc., 248(2):269–309, 1979

    work page 1979

  23. [23]

    M. A. Campercholi. Dominions and primitive positive functions.Journal of Symbolic Logic, 83(1):40–54, 2018

    work page 2018

  24. [24]

    M. A. Campercholi and J. G. Raftery. Relative congruence formulas and decompositions in quasivarieties. Algebra Universalis, 78(3):407–425, 2017

    work page 2017

  25. [25]

    M. A. Campercholi and D. J. Vaggione. Implicit definition of the quaternary discriminator.Algebra Universalis, 68(1-2):1–16, 2012

    work page 2012

  26. [26]

    M. A. Campercholi and D. J. Vaggione. Semantical conditions for the definability of functions and relations.Algebra Universalis, 76:71–98, 2015

    work page 2015

  27. [27]

    The theory of implicit operations

    L. Carai, M. Kurtzhals, and T. Moraschini. An addendum to “The theory of implicit operations”. Available on arXiv, 2025

    work page 2025

  28. [28]

    Carai, M

    L. Carai, M. Kurtzhals, and T. Moraschini. Congruence permutability in quasivarieties. Available at arXiv:2512.09064, 2025

    work page arXiv 2025

  29. [29]

    Carai, M

    L. Carai, M. Kurtzhals, and T. Moraschini. Epimorphisms between finitely generated algebras.Indag. Math. (N.S.), 36(5):1336–1354, 2025

    work page 2025

  30. [30]

    Carai, M

    L. Carai, M. Kurtzhals, and T. Moraschini. Implicit operations in reduced commutative rings.Manuscript,

    work page

  31. [31]

    Carai, M

    L. Carai, M. Kurtzhals, and T. Moraschini. Implicit operations in varieties of commutative monoids. In preparation, 2025

    work page 2025

  32. [32]

    Casta˜ no, M

    D. Casta˜ no, M. Campercholi, and D. Vaggione. Preservation theorems for AE-sentences.Journal of Symbolic Logic, pages 1–17, 2025

    work page 2025

  33. [33]

    Chagrov and M

    A. Chagrov and M. Zakharyaschev.Modal logic, volume 35 ofOxford Logic Guides. The Clarendon Press, Oxford University Press, New York, 1997. Oxford Science Publications. 118 LUCA CARAI, MIRIAM KURTZHALS, AND TOMMASO MORASCHINI

    work page 1997

  34. [34]

    C. C. Chang. A new proof of the completeness of the Lukasiewicz axioms.Trans. Amer. Math. Soc., 93:74–80, 1959

    work page 1959

  35. [35]

    C. C. Chang and H. J. Keisler.Model theory, volume 73 ofStudies in Logic and the Foundations of Mathematics. North-Holland Publishing Co., Amsterdam, third edition, 1990

    work page 1990

  36. [36]

    R. L. O. Cignoli, I. M. L. D’Ottaviano, and D. Mundici.Algebraic foundations of many-valued reasoning, volume 7 ofTrends in Logic—Studia Logica Library. Kluwer Academic Publishers, Dordrecht, 2000

    work page 2000

  37. [37]

    G. T. Clarke. Semigroup varieties with the amalgamation property.J. Algebra, 80(1):60–72, 1983

    work page 1983

  38. [38]

    A. H. Clifford and G. B. Preston.The algebraic theory of semigroups. Vol. I. Mathematical Surveys, No. 7. American Mathematical Society, Providence, RI, 1961

    work page 1961

  39. [39]

    A. H. Clifford and G. B. Preston.The algebraic theory of semigroups. Vol. II. Mathematical Surveys, No. 7. American Mathematical Society, Providence, RI, 1967

    work page 1967

  40. [40]

    Czelakowski and W

    L. Czelakowski and W. Dziobiak. Congruence distributive quasivarieties whose finitely subdirectly irreducible members form a universal class.Algebra Universalis, 27(1):128–149, 1990

    work page 1990

  41. [41]

    B. A. Davey and H. Werner. Dualities and equivalences for varieties of algebras. InContributions to lattice theory (Szeged, 1980), volume 33 ofColloq. Math. Soc. J´ anos Bolyai, pages 101–275. North-Holland, Amsterdam, 1983

    work page 1980

  42. [42]

    P. M. Dekker.Interpolation and Beth definability in implicative fragments of IPC. PhD thesis, University of Amsterdam, 2020

    work page 2020

  43. [43]

    Di Nola and A

    A. Di Nola and A. Lettieri. One chain generated varieties of MV-algebras.J. Algebra, 225(2):667–697, 2000

    work page 2000

  44. [44]

    Diego.Sobre ´ algebras de Hilbert

    A. Diego.Sobre ´ algebras de Hilbert. PhD thesis, Universidad de Buenos Aires, 1961. Available at http://digital.bl.fcen.uba.ar/Download/Tesis/Tesis_1092_Diego.pdf

    work page 1961

  45. [45]

    D. S. Dummit and R. M. Foote.Abstract algebra. John Wiley & Sons, Inc., Hoboken, NJ, third edition, 2004

    work page 2004

  46. [46]

    J. M. Dunn and R. K. Meyer. Algebraic completeness results for Dummett’s LC and its extensions.Z. Math. Logik Grundlagen Math., 17:225–230, 1971

    work page 1971

  47. [47]

    Dvureˇ censkij and O

    A. Dvureˇ censkij and O. Zahiri. On epicompleteM V-algebras.J. Appl. Logics, 5(1):165–183, 2018

    work page 2018

  48. [48]

    Eisenbud.Commutative algebra, volume 150 ofGraduate Texts in Mathematics

    D. Eisenbud.Commutative algebra, volume 150 ofGraduate Texts in Mathematics. Springer-Verlag, New York, 1995. With a view toward algebraic geometry

    work page 1995

  49. [49]

    A. L. Foster. Generalized “Boolean” theory of universal algebras. I. Subdirect sums and normal representation theorem.Math. Z., 58:306–336, 1953

    work page 1953

  50. [50]

    A. L. Foster. Generalized “Boolean” theory of universal algebras. II. Identities and subdirect sums of functionally complete algebras.Math. Z., 59:191–199, 1953

    work page 1953

  51. [51]

    Fried, G

    E. Fried, G. Gr¨ atzer, and R. Quackenbush. Uniform congruence schemes.Algebra Universalis, 10(2):176– 188, 1980

    work page 1980

  52. [52]

    Fried and E

    E. Fried and E. W. Kiss. Connections between congruence-lattices and polynomial properties.Algebra Universalis, 17(3):227–262, 1983

    work page 1983

  53. [53]

    Gerla.Many-Valued Logics of Continuous t-Norms and Their Functional Representation

    B. Gerla.Many-Valued Logics of Continuous t-Norms and Their Functional Representation. Ph.D. thesis, University of Milan, 2001. Available at:https://www.dicom.uninsubria.it/ ~bgerla/tesi.pdf

    work page 2001

  54. [54]

    B. Gerla. Rational Lukasiewicz logic and divisible MV-algebras.Neural Networks World, 10, 2001

    work page 2001

  55. [55]

    Gispert and D

    J. Gispert and D. Mundici. MV-algebras: a variety for magnitudes with Archimedean units.Algebra Universalis, 53(1):7–43, 2005

    work page 2005

  56. [56]

    Gispert and A

    J. Gispert and A. Torrens. Quasivarieties generated by simple MV-algebras. volume 61, pages 79–99

    work page

  57. [57]

    Gispert and A

    J. Gispert and A. Torrens. Quasivarieties generated by simple MV-algebras.Studia Logica, 61(1):79–99, 1998

    work page 1998

  58. [58]

    V. A. Gorbunov.Algebraic theory of quasivarieties. Siberian School of Algebra and Logic. Consultants Bureau, New York, 1998. Translated from the Russian

    work page 1998

  59. [59]

    Gr¨ atzer.Universal algebra

    G. Gr¨ atzer.Universal algebra. Springer, New York, second edition, 2008. THE THEORY OF IMPLICIT OPERATIONS 119

    work page 2008

  60. [60]

    Gurevich and A

    Y. Gurevich and A. I. Kokorin. Universal equivalence of ordered Abelian groups.Algebra i Logika Sem., 2(1):37–39, 1963

    work page 1963

  61. [61]

    Kluwer Academic Publishers, Dordrecht, 1998

    Petr H´ ajek.Metamathematics of fuzzy logic, volume 4 ofTrends in Logic—Studia Logica Library. Kluwer Academic Publishers, Dordrecht, 1998

    work page 1998

  62. [62]

    Hecht and T

    T. Hecht and T. Katriˇ n´ ak. Equational classes of relative Stone algebras.Notre Dame J. Formal Log., 13:248–254, 1972

    work page 1972

  63. [63]

    Hodges.Model Theory

    W. Hodges.Model Theory. Cambridge University Press, 1993

    work page 1993

  64. [64]

    Hoogland

    E. Hoogland. Algebraic characterizations of various Beth definability properties.Studia Logica, 65(1):91– 112, 2000

    work page 2000

  65. [65]

    Hoogland.Definability and Interpolation: Model -theoretic Investigations

    E. Hoogland.Definability and Interpolation: Model -theoretic Investigations. Ph.D. thesis, University of Amsterdam, 2001

    work page 2001

  66. [66]

    A. Horn. The separation theorem of intuitionist propositional calculus.J. Symbolic Logic, 27:391–399, 1962

    work page 1962

  67. [67]

    A. Horn. Logic with truth values in a linearly ordered Heyting algebra.J. Symb. Logic, 34:395–408, 1969

    work page 1969

  68. [68]

    J. M. Howie. Isbell’s zigzag theorem and its consequences. InSemigroup theory and its applications (New Orleans, LA, 1994), volume 231 ofLondon Math. Soc. Lecture Note Ser., pages 81–91. Cambridge Univ. Press, Cambridge, 1996

    work page 1994

  69. [69]

    J. M. Howie and J. R. Isbell. Epimorphisms and dominions. II.J. Algebra, 6:7–21, 1967

    work page 1967

  70. [70]

    T. Hu. Stone duality for primal algebra theory.Math. Z., 110:180–198, 1969

    work page 1969

  71. [71]

    J. R. Isbell. Epimorphisms and dominions. InProc. Conf. Categorical Algebra (La Jolla, Calif., 1965), pages 232–246. Springer-Verlag New York, Inc., New York, 1966

    work page 1965

  72. [72]

    J. R. Isbell. Epimorphisms and dominions. IV.J. London Math. Soc. (2), 1:265–273, 1969

    work page 1969

  73. [73]

    Kaarli and A

    K. Kaarli and A. F. Pixley.Polynomial completeness in algebraic systems. Chapman & Hall/CRC, Boca Raton, FL, 2001

    work page 2001

  74. [74]

    K. A. Kearnes and E. W. Kiss. The shape of congruence lattices.Mem. Amer. Math. Soc., 222(1046), 2013

    work page 2013

  75. [75]

    Kimura.On Semigroups

    N. Kimura.On Semigroups. Ph.D. thesis, Tulane University of Louisiana, 1957

    work page 1957

  76. [76]

    E. W. Kiss, L. M´ arki, P. Pr¨ ohle, and W. Tholen. Categorical algebraic properties. A compendium on amalgamation, congruence extension, epimorphisms, residual smallness, and injectivity.Studia Sci. Math. Hungar., 18(1):79–140, 1982

    work page 1982

  77. [77]

    K¨ ohler

    P. K¨ ohler. Brouwerian semilattices.Trans. Amer. Math. Soc., 268(1):103–126, 1981

    work page 1981

  78. [78]

    K¨ ohler and D

    P. K¨ ohler and D. Pigozzi. Varieties with equationally definable principal congruences.Algebra Universalis, 11(2):213–219, 1980

    work page 1980

  79. [79]

    Y. Komori. The finite model property of the intermediate propositional logics on finite slices.J. Fac. Sci. Univ. Tokyo Sect. IA Math., 22(2):117–120, 1975

    work page 1975

  80. [80]

    V. M. Kopytov and N. Ya. Medvedev.The theory of lattice-ordered groups, volume 307 ofMathematics and its Applications. Kluwer Academic Publishers Group, Dordrecht, 1994

    work page 1994

Showing first 80 references.