Cofibrant generation of pure monomorphisms in presheaf categories

Ji\v{r}\'i Rosick\'y; Jonathan Feigert; Marcos Mazari-Armida; Mark Kamsma; Sean Cox

arxiv: 2506.20278 · v3 · submitted 2025-06-25 · 🧮 math.CT · math.LO

Cofibrant generation of pure monomorphisms in presheaf categories

Sean Cox , Jonathan Feigert , Mark Kamsma , Marcos Mazari-Armida , Ji\v{r}\'i Rosick\'y This is my paper

Pith reviewed 2026-05-19 08:27 UTC · model grok-4.3

classification 🧮 math.CT math.LO

keywords cofibrant generationpure monomorphismspresheaf categoriesstable independence relationmonoid actionscategory of acts

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The pith

Pure monomorphisms in presheaf categories are cofibrantly generated precisely when the base category C satisfies a specific divisibility condition.

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

The paper establishes a characterization of when pure monomorphisms form a cofibrantly generated class in presheaf categories. This depends on whether the small category C has the property that any two morphisms can be related through composition with a third in a certain way. For the case of a monoid, it reduces to a concrete algebraic condition on its elements. The authors use model theory to link this to stable independence relations. They apply the result to show that pure monomorphisms over the natural numbers monoid are not cofibrantly generated.

Core claim

We characterise when the pure monomorphisms in a presheaf category Set^C are cofibrantly generated in terms of the category C. In particular, when C is a monoid S this characterises cofibrant generation of pure monomorphisms between sets with an S-action in terms of S: this happens if and only if for all a, b ∈ S there is c ∈ S such that a = cb or ca = b. We give a model-theoretic proof: we prove that our characterisation is equivalent to having a stable independence relation, which in turn is equivalent to cofibrant generation. As a corollary, we show that pure monomorphisms in acts over the multiplicative monoid of natural numbers are not cofibrantly generated.

What carries the argument

The equivalence chain connecting a divisibility condition on the category C to the existence of a stable independence relation and hence to cofibrant generation of the pure monomorphisms.

If this is right

If the condition holds for C, then the pure monomorphisms in Set^C are cofibrantly generated by some set of maps.
For a monoid S satisfying the condition, pure monomorphisms in the category of S-acts are cofibrantly generated.
The multiplicative monoid of natural numbers does not satisfy the condition, hence its pure monomorphisms are not cofibrantly generated.

Where Pith is reading between the lines

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

The condition resembles left or right divisibility, which might relate to known properties like being a group or having common multiples.
This approach could be used to check the property for other concrete monoids such as the integers or free monoids.
Model-theoretic techniques might help identify similar generation properties for other morphism classes in presheaf categories.

Load-bearing premise

The assumption that the combinatorial condition on C is equivalent to the existence of a stable independence relation, which is equivalent to cofibrant generation of pure monomorphisms.

What would settle it

Finding a category C that satisfies the divisibility condition but whose pure monomorphisms are not cofibrantly generated, or vice versa.

read the original abstract

We characterise when the pure monomorphisms in a presheaf category $\mathbf{Set}^\mathcal{C}$ are cofibrantly generated in terms of the category $\mathcal{C}$. In particular, when $\mathcal{C}$ is a monoid $S$ this characterises cofibrant generation of pure monomorphisms between sets with an $S$-action in terms of $S$: this happens if and only if for all $a, b \in S$ there is $c \in S$ such that $a = cb$ or $ca = b$. We give a model-theoretic proof: we prove that our characterisation is equivalent to having a stable independence relation, which in turn is equivalent to cofibrant generation. As a corollary, we show that pure monomorphisms in acts over the multiplicative monoid of natural numbers are not cofibrantly generated.

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 pins down a concrete combinatorial condition on a monoid that decides whether pure monomorphisms in the category of S-acts are cofibrantly generated.

read the letter

The main point is that this paper gives a simple combinatorial test for when the pure monomorphisms in a presheaf category are cofibrantly generated. For the case where the category is a monoid S, it boils down to: for all a, b in S there exists c in S with a = c b or c a = b. They prove this is equivalent to the existence of a stable independence relation, which itself is equivalent to cofibrant generation. As a corollary they show the condition fails for the multiplicative monoid of natural numbers, so pure monomorphisms there are not cofibrantly generated. What is new is the explicit if-and-only-if for monoids together with the link to stable independence from model theory. The model-theoretic proof organizes some previously separate combinatorial and model-theoretic notions in a useful way. The paper does well at stating the result clearly in terms of the base category C and at giving a concrete application as the corollary. The approach stays focused on the equivalences without unnecessary extra machinery. The potential soft spot is in the equivalence chain itself. The argument rests on the combinatorial condition implying a stable independence relation and that relation implying cofibrant generation. If the general theorems they cite apply directly in the presheaf or act setting, with the required hypotheses on exactness and stability verified, then the characterization holds. One would want to check the details of how the independence relation is built and shown to be stable for this specific case. The stress-test note correctly flags this as the least secure step, but the paper appears to address it through the model-theoretic steps rather than leaving it tacit. This work is for specialists in categorical model theory or those building model structures on categories of acts and presheaves. A reader looking for an explicit criterion that ties cofibrant generation to a simple condition on the monoid would get direct value from it. It shows clear thinking by connecting the combinatorial property to the independence relation without circularity or invented entities. I would bring it to a reading group with people working in these areas. It deserves peer review because the characterization is new, the equivalences are laid out, and the corollary gives a falsifiable application.

Referee Report

1 major / 1 minor

Summary. The paper characterizes when the pure monomorphisms in a presheaf category Set^C are cofibrantly generated, in terms of the category C. For C a monoid S, this holds if and only if for all a, b in S there exists c in S such that a = c b or c a = b. The proof is model-theoretic: the combinatorial condition is shown equivalent to the existence of a stable independence relation on the presheaves (or S-acts), which is in turn equivalent to cofibrant generation of the pure monomorphisms. A corollary establishes that pure monomorphisms in acts over the multiplicative monoid of natural numbers are not cofibrantly generated.

Significance. If the equivalences are fully verified, the result supplies a concrete, checkable combinatorial criterion for cofibrant generation of pure monomorphisms in presheaf categories, linking accessible-category techniques with model-theoretic independence relations. The explicit monoid condition and the negative corollary for the natural-numbers monoid are particularly useful for applications in algebra and model theory.

major comments (1)

[Abstract and model-theoretic proof section] Abstract and the section establishing the equivalence chain: the central if-and-only-if characterization rests on the two-step equivalence (combinatorial condition on C ⇔ existence of a stable independence relation ⇔ cofibrant generation of pure monomorphisms). The manuscript invokes general results from model theory and accessible categories for the second step but does not explicitly confirm that the independence relation constructed for presheaves (or S-acts) satisfies all required hypotheses (exactness, stability, extension property) of the cited theorem; this verification is load-bearing for the claimed equivalence.

minor comments (1)

[Introduction] The definition of pure monomorphisms and the precise statement of the stable independence relation would benefit from an explicit example for a small monoid early in the paper to aid readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying the need for more explicit verification in the equivalence chain. We have revised the paper to strengthen this part of the argument.

read point-by-point responses

Referee: [Abstract and model-theoretic proof section] Abstract and the section establishing the equivalence chain: the central if-and-only-if characterization rests on the two-step equivalence (combinatorial condition on C ⇔ existence of a stable independence relation ⇔ cofibrant generation of pure monomorphisms). The manuscript invokes general results from model theory and accessible categories for the second step but does not explicitly confirm that the independence relation constructed for presheaves (or S-acts) satisfies all required hypotheses (exactness, stability, extension property) of the cited theorem; this verification is load-bearing for the claimed equivalence.

Authors: We agree that the manuscript would benefit from a more explicit verification that the constructed independence relation satisfies the full set of hypotheses (exactness, stability, and the extension property) required by the general theorem. In the revised version we have added a dedicated paragraph in the model-theoretic section that directly checks each hypothesis for the relation on presheaves (and, in the monoid case, on S-acts), using the combinatorial condition on C to establish the required properties. The abstract itself requires no change, as it accurately summarises the overall strategy. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation relies on independent model-theoretic equivalences

full rationale

The paper establishes a direct combinatorial characterization of cofibrant generation for pure monomorphisms in presheaf categories Set^C, with the monoid case reducing to the explicit condition ∀a,b∈S ∃c∈S (a=cb ∨ ca=b). The model-theoretic proof shows equivalence of this condition to the existence of a stable independence relation and then invokes the general link from such relations to cofibrant generation; neither step is a self-definition, fitted parameter renamed as prediction, or reduction by construction to the paper's own inputs. The cited equivalences draw on external results in accessible categories and model theory rather than prior self-citations that bear the full load of the central claim, rendering the derivation self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard background axioms of category theory for presheaves and model theory for independence relations; no free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (2)

standard math Standard axioms of category theory and the definition of pure monomorphisms in presheaf categories
The paper invokes these as background to define the objects of study.
domain assumption Existence and properties of stable independence relations in the relevant categories
The equivalence proof relies on this model-theoretic concept being well-defined and behaving as stated.

pith-pipeline@v0.9.0 · 5694 in / 1562 out tokens · 58810 ms · 2026-05-19T08:27:30.097368+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 5.1: pure monomorphisms in Set^C cofibrantly generated iff C locally linearly preordered iff stable independence relation exists on Set^C_pure

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages

[1]

Ad´ amek, H

J. Ad´ amek, H. Herrlich, J. Rosick´ y and W. Tholen, Weak Factorization Systems and Topological Functors, Applied Categorical Structures 10 (2002), 237-249

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Ad´ amek and J

J. Ad´ amek and J. Rosick´ y,Locally Presentable and Accessible Categories , Cambridge Uni- versity Press (1994)

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Ad´ amek and J

J. Ad´ amek and J. Rosick´ y,On Pure Quotients and Pure Subobjects , Czechoslovak Mathe- matical Journal 54 (2004), 623-636. COFIBRANT GENERATION OF PURE MONOMORPHISMS IN PRESHEAF CAT EGORIES 23

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B. Banaschewski and E. M. Nelson, Equational compactness in equational classes of alge- bras, Algebra Universalis 2 (1972), 152-165

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Kamsma and J

M. Kamsma and J. Rosick´ y, Lifting independence along functors , preprint, http://arxiv.org/abs/2411.14813 (2024)

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M. Kilp, U. Knauer, A. V. Mikhalev, Monoids, Acts and Categories , De Gruyter (2011)

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Lieberman, L

M. Lieberman, L. Positselski, J. Rosick´ y and S. Vasey, Coﬁbrant generation of pure monomorphisms, Journal of Algebra 560 (2020), 1297-1310

work page 2020
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Lieberman, J

M. Lieberman, J. Rosick´ y and S. Vasey, Forking independence from the categorical point of view , Advances in Mathematics 346 (2019), 719-772

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Lieberman, J

M. Lieberman, J. Rosick´ y and S. Vasey, Cellular categories and stable independence , The Journal of Symbolic Logic 88, (2023)

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Mazari-Armida and J

M. Mazari-Armida and J. Rosick´ y,Relative injective modules, superstability and noetheria n categories, Journal of Mathematical Logic, online (2024)

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T. G. Mustaﬁn, Stability of the theory of polygons , Proceedings of the Institute of Mathe- matics 8 (1988), 92-108 (in Russian); translated in Model Theory a nd Applications, Amer- ican Mathematical Society translations 295 (1999), 205-223

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Makkai and R

M. Makkai and R. Par´ e, Accessible Categories: The Foundation of Categorical Mode l The- ory, Contemporary Mathematics 104 (1989)

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Prest, Purity, Spectra and Localisation , Cambridge University Press (2009)

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Pr¨ ufer, Untersuchungen ¨ uber die Zerlegbarkeit der abz¨ ahlbaren p rim¨ aren Abelschen Gruppen, Mathematische Zeitschrift 17 (1923), 35–61

H. Pr¨ ufer, Untersuchungen ¨ uber die Zerlegbarkeit der abz¨ ahlbaren p rim¨ aren Abelschen Gruppen, Mathematische Zeitschrift 17 (1923), 35–61

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Rothmaler, Purity in model theory , in Advances in algebra and model theory (ed

P. Rothmaler, Purity in model theory , in Advances in algebra and model theory (ed. by M. Droste and R. G¨ obel), Gordon and Breach 1997, 445–470

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ˇSaroch and J

J. ˇSaroch and J. Trlifaj, Test sets for factorization properties of modules , Rendiconti del Seminario Matematico della Universit` a di Padova 144 (2020), 217-2 38

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Shelah, Classiﬁcation theory and the number of nonisomorphic model s, North-Holland Publishing (1990)

S. Shelah, Classiﬁcation theory and the number of nonisomorphic model s, North-Holland Publishing (1990)

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Taylor, Residually small varieties , Algebra Universalis 2 (1972), 33-53

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Taylor, Pure-irreducible mono-unary algebras , Algebra Universalis (1974), 235-243

W. Taylor, Pure-irreducible mono-unary algebras , Algebra Universalis (1974), 235-243

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[27]

Taylor, Pure compactiﬁcations in quasi-primal varieties , Canadian Journal of Mathe- matics 28 (1976), 50-62

W. Taylor, Pure compactiﬁcations in quasi-primal varieties , Canadian Journal of Mathe- matics 28 (1976), 50-62

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[28]

Tent and M

K. Tent and M. Ziegler, A Course in Model Theory , Cambridge University Press (2012)

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[29]

Wenzel Equational compactness, in: G

G.H. Wenzel Equational compactness, in: G. Gr¨ atzer, Universal Algebra, Springer-Verlag (1979). 24 S. COX, J. FEIGERT, M. KAMSMA, M. MAZARI-ARMIDA AND J. ROSI CK ´Y

work page 1979
[30]

Wenzel, Subdirect irreducibility and equational compactness in un ary algebras, Archiv der Mathematik 21 (1970), 256-264

G.H. Wenzel, Subdirect irreducibility and equational compactness in un ary algebras, Archiv der Mathematik 21 (1970), 256-264. Sean Cox Department of Mathematics and Applied Mathematics Virginia Commonwealth University 1015 Floyd A ve, V A 23284, Richmond, USA Email address : scox9@vcu.edu Jonathan Feigert Department of Mathematics Baylor University W aco...

work page 1970

[1] [1]

Ad´ amek, H

J. Ad´ amek, H. Herrlich, J. Rosick´ y and W. Tholen, Weak Factorization Systems and Topological Functors, Applied Categorical Structures 10 (2002), 237-249

work page 2002

[2] [2]

Ad´ amek and J

J. Ad´ amek and J. Rosick´ y,Locally Presentable and Accessible Categories , Cambridge Uni- versity Press (1994)

work page 1994

[3] [3]

Ad´ amek and J

J. Ad´ amek and J. Rosick´ y,On Pure Quotients and Pure Subobjects , Czechoslovak Mathe- matical Journal 54 (2004), 623-636. COFIBRANT GENERATION OF PURE MONOMORPHISMS IN PRESHEAF CAT EGORIES 23

work page 2004

[4] [4]

Banaschewski, Equational Compactness of G-Sets , Canadian Mathematical Bulletin 17 (1974), 11-18

B. Banaschewski, Equational Compactness of G-Sets , Canadian Mathematical Bulletin 17 (1974), 11-18

work page 1974

[5] [5]

Banaschewski and G

B. Banaschewski and G. Bruns, Categorical characterization of the MacNeille completion , Archiv der Mathematik 18 (1967), 369-377

work page 1967

[6] [6]

Banaschewski and E

B. Banaschewski and E. M. Nelson, Equational compactness in equational classes of alge- bras, Algebra Universalis 2 (1972), 152-165

work page 1972

[7] [7]

Barr, On categories with eﬀective unions , Categorical Algebra and its Applications, Lecture Notes in Mathematics 1348, Springer-Verlag (1988), 19– 35

M. Barr, On categories with eﬀective unions , Categorical Algebra and its Applications, Lecture Notes in Mathematics 1348, Springer-Verlag (1988), 19– 35

work page 1988

[8] [8]

Beke, Sheaﬁﬁable homotopy model categories , Mathematical Proceedings of the Cam- bridge Philosophical Society 129 (2000), 447-475

T. Beke, Sheaﬁﬁable homotopy model categories , Mathematical Proceedings of the Cam- bridge Philosophical Society 129 (2000), 447-475

work page 2000

[9] [9]

Borceux and J

F. Borceux and J. Rosick´ y, Purity in algebra , Algebra Universalis 56 (2007), 17-35

work page 2007

[10] [10]

Joyal and R

A. Joyal and R. Street, Pullbacks equivalent to pseudopullbacks , Cahiers de Topologie et G´ eom´ etrie Diﬀ´ erentielle Cat´ egoriques 34 (1993), 153-156

work page 1993

[11] [11]

Kamsma, NSOP1-like independence in AECats , The Journal of Symbolic Logic 89 (2024), 724-757

M. Kamsma, NSOP1-like independence in AECats , The Journal of Symbolic Logic 89 (2024), 724-757

work page 2024

[12] [12]

Kamsma and J

M. Kamsma and J. Rosick´ y, Lifting independence along functors , preprint, http://arxiv.org/abs/2411.14813 (2024)

work page arXiv 2024

[13] [13]

M. Kilp, U. Knauer, A. V. Mikhalev, Monoids, Acts and Categories , De Gruyter (2011)

work page 2011

[14] [14]

Lieberman, L

M. Lieberman, L. Positselski, J. Rosick´ y and S. Vasey, Coﬁbrant generation of pure monomorphisms, Journal of Algebra 560 (2020), 1297-1310

work page 2020

[15] [15]

Lieberman, J

M. Lieberman, J. Rosick´ y and S. Vasey, Forking independence from the categorical point of view , Advances in Mathematics 346 (2019), 719-772

work page 2019

[16] [16]

Lieberman, J

M. Lieberman, J. Rosick´ y and S. Vasey, Cellular categories and stable independence , The Journal of Symbolic Logic 88, (2023)

work page 2023

[17] [17]

Mazari-Armida and J

M. Mazari-Armida and J. Rosick´ y,Relative injective modules, superstability and noetheria n categories, Journal of Mathematical Logic, online (2024)

work page 2024

[18] [18]

T. G. Mustaﬁn, Stability of the theory of polygons , Proceedings of the Institute of Mathe- matics 8 (1988), 92-108 (in Russian); translated in Model Theory a nd Applications, Amer- ican Mathematical Society translations 295 (1999), 205-223

work page 1988

[19] [19]

Makkai and R

M. Makkai and R. Par´ e, Accessible Categories: The Foundation of Categorical Mode l The- ory, Contemporary Mathematics 104 (1989)

work page 1989

[20] [20]

Prest, Purity, Spectra and Localisation , Cambridge University Press (2009)

M. Prest, Purity, Spectra and Localisation , Cambridge University Press (2009)

work page 2009

[21] [21]

Pr¨ ufer, Untersuchungen ¨ uber die Zerlegbarkeit der abz¨ ahlbaren p rim¨ aren Abelschen Gruppen, Mathematische Zeitschrift 17 (1923), 35–61

H. Pr¨ ufer, Untersuchungen ¨ uber die Zerlegbarkeit der abz¨ ahlbaren p rim¨ aren Abelschen Gruppen, Mathematische Zeitschrift 17 (1923), 35–61

work page 1923

[22] [22]

Rothmaler, Purity in model theory , in Advances in algebra and model theory (ed

P. Rothmaler, Purity in model theory , in Advances in algebra and model theory (ed. by M. Droste and R. G¨ obel), Gordon and Breach 1997, 445–470

work page 1997

[23] [23]

ˇSaroch and J

J. ˇSaroch and J. Trlifaj, Test sets for factorization properties of modules , Rendiconti del Seminario Matematico della Universit` a di Padova 144 (2020), 217-2 38

work page 2020

[24] [24]

Shelah, Classiﬁcation theory and the number of nonisomorphic model s, North-Holland Publishing (1990)

S. Shelah, Classiﬁcation theory and the number of nonisomorphic model s, North-Holland Publishing (1990)

work page 1990

[25] [25]

Taylor, Residually small varieties , Algebra Universalis 2 (1972), 33-53

W. Taylor, Residually small varieties , Algebra Universalis 2 (1972), 33-53

work page 1972

[26] [26]

Taylor, Pure-irreducible mono-unary algebras , Algebra Universalis (1974), 235-243

W. Taylor, Pure-irreducible mono-unary algebras , Algebra Universalis (1974), 235-243

work page 1974

[27] [27]

Taylor, Pure compactiﬁcations in quasi-primal varieties , Canadian Journal of Mathe- matics 28 (1976), 50-62

W. Taylor, Pure compactiﬁcations in quasi-primal varieties , Canadian Journal of Mathe- matics 28 (1976), 50-62

work page 1976

[28] [28]

Tent and M

K. Tent and M. Ziegler, A Course in Model Theory , Cambridge University Press (2012)

work page 2012

[29] [29]

Wenzel Equational compactness, in: G

G.H. Wenzel Equational compactness, in: G. Gr¨ atzer, Universal Algebra, Springer-Verlag (1979). 24 S. COX, J. FEIGERT, M. KAMSMA, M. MAZARI-ARMIDA AND J. ROSI CK ´Y

work page 1979

[30] [30]

Wenzel, Subdirect irreducibility and equational compactness in un ary algebras, Archiv der Mathematik 21 (1970), 256-264

G.H. Wenzel, Subdirect irreducibility and equational compactness in un ary algebras, Archiv der Mathematik 21 (1970), 256-264. Sean Cox Department of Mathematics and Applied Mathematics Virginia Commonwealth University 1015 Floyd A ve, V A 23284, Richmond, USA Email address : scox9@vcu.edu Jonathan Feigert Department of Mathematics Baylor University W aco...

work page 1970