Conjugate L-Subgroups of an L-group and their Applications to Normality and Normalizer

Ananya Manas; Iffat Jahan

REVIEW 3 major objections 5 minor 1 cited by

Conjugate L-Subgroups of an L-group and their Applications to Normality and Normalizer

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read For L-subgroups, conjugation by an L-point is compatible with the normalizer: $N(\eta)^{a_z}=a\wedge N(\eta^{a_z})$.

desk verdict A genuinely new conjugate-by-L-point operation in L-group theory with a solid central normalizer theorem, but the paper states its foundational definition one way and uses it the other; fix that convention and it is ready for serious review. read the letter →

arxiv 2506.20692 v1 pith:IIMCR42X submitted 2025-06-25 math.GR math.RA

Iffat Jahan , Ananya Manas This is my paper · ORCID

classification math.GRmath.RA MSC 20N2506D10

keywords L-algebraL-subgroupL-groupconjugatenormalnormalizerL-pointcompletelydistributivelattice

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper introduces a notion of conjugation for L-subgroups of an L-group, i.e. lattice-valued fuzzy subgroups of a lattice-valued fuzzy group. Instead of conjugating by an element of the underlying ordinary group, it conjugates by an L-point of the parent L-group, giving the conjugate $\eta^{a_z}(x)=a\wedge\eta(zxz^{-1})$. The authors show this conjugate is again an L-subgroup and behaves like classical group conjugation with respect to products, homomorphic images and preimages, level subsets, and (on chain-valued lattices) maximality. They then establish that normal L-subgroups are exactly those that contain every conjugate, and prove the central identity $N(\eta)^{a_z}(g)=a\wedge N(\eta^{a_z})(g)$, which allows the normalizer to be redefined as the join of all L-points whose conjugation leaves the L-subgroup contained in itself. The upshot is a lattice-valued analogue of the classical fact that the normalizer is the set of elements whose conjugates stay inside the subgroup.

What carries the argument

The central object is the L-point $a_z\in\mu$, a lattice-valued singleton that takes value $a$ at $z$ and $0$ elsewhere, and the conjugate L-subgroup $\eta^{a_z}(x)=a\wedge\eta(zxz^{-1})$ built from it. This conjugate is defined by specializing the earlier conjugate-by-an-L-subset operation of [11] to L-points. The load-bearing identity is Theorem 4.4, $N(\eta)^{a_z}(g)=a\wedge N(\eta^{a_z})(g)$, and the mechanism that carries it is the coset equality $b_g\circ\eta=\eta\circ b_g$: an L-point $b_g$ belongs to the normalizer exactly when its left and right cosets of $\eta$ coincide. Complete distributivity of $L$ is then used to pull the meet with $a$ through the join over all such $b$, which is the step on which the identity depends.

What would settle it

Compute both sides of $N(\eta)^{a_z}(g)=a\wedge N(\eta^{a_z})(g)$ for a small finite group $G$, an L-subgroup $\eta$ with simple level subgroups, and an L-point $a_z$, using as $L$ a complete lattice that is not completely distributive, for example the open subsets of the real line. If any choice of $g$ gives different values on the two sides, Theorem 4.4 fails as stated; if equality holds for every such choice, the complete distributivity assumption can be weakened.

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Extended reading notes

Core claim

The central claim is that conjugacy and normalizer formation commute in the L-setting up to the value of the conjugating L-point. Concretely, for every L-subgroup $\eta$ of an L-group $\mu$, every L-point $a_z$ of $\mu$, and every $g\in G$, $N(\eta)^{a_z}(g)=a\wedge N(\eta^{a_z})(g)$. The paper proves this by rewriting both sides through the coset condition $b_g\circ\eta=\eta\circ b_g$ that defines membership in the normalizer, then using complete distributivity of the lattice $L$ to pass the meet with $a$ through the join over the defining levels $b$. The identity supports a new definition of the normalizer, $N(\eta)=\bigcup\{a_z\in\mu\mid \eta^{a_z}\subseteq\eta\}$, which directly mirrors the classical characterization of the normalizer as the set of elements $x$ with $H^x\subseteq H$. Along the way the paper proves that $\eta$ is a normal L-subgroup of $\mu$ if and only if $\eta^{a_z}\subseteq\eta$ for every L-point $a_z$ of $\mu$.

Load-bearing premise

The load-bearing premise is that the lattice $L$ of truth values is completely distributive, because the proof of Theorem 4.4 moves a meet through an arbitrary join; if $L$ has only ordinary distributivity, the central identity is not established.

Editorial extensions

If this is right

Normal L-subgroups are characterized by conjugate containment: $\eta$ is normal in $\mu$ if and only if $\eta^{a_z}\subseteq\eta$ for every L-point $a_z$ of $\mu$; when the tips coincide, every conjugate equals $\eta$ itself.
Because $N(\eta)$ is the largest L-subgroup of $\mu$ in which $\eta$ is normal, the identity $N(\eta)^{a_z}=a\wedge N(\eta^{a_z})$ transfers normalizers across conjugation: the conjugate of a normalizer is the normalizer of the conjugate, up to the value $a$.
The level-set characterization of Theorem 3.6 reduces conjugate L-subgroups to ordinary subgroup conjugacy on each level: $\nu=\eta^{a_z}$ if and only if $\nu_t=\eta_t^{z^{-1}}$ for every $t$ at most the tip of $\nu$.
The characteristic-function version recovers the classical statement: two ordinary subgroups $H$ and $K$ are conjugate in $G$ exactly when $1_K$ is conjugate to $1_H$ as L-subgroups of $1_G$, and their normalizers satisfy $N(1_H)^{1_x}=N((1_H)^{1_x})$.
On a chain-valued lattice, conjugating a maximal L-subgroup by an L-point yields either the whole conjugate parent or again a maximal L-subgroup, giving a conjugate version of maximality.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper's motivation names pronormal and abnormal subgroups as targets for the L-setting; the compatibility identity proved here is exactly the kind of bridge that would be needed to define and test those notions.
Complete distributivity is used in only one step in Theorem 4.4, exchanging a meet with a join over the normalizer levels, so a natural testable extension is whether the identity survives on complete lattices satisfying only that specific distributive law, or on finite lattices where the law is automatic.
Since the normalizer identity holds for L-points, one could investigate an L-valued version of the classical theorem that a subgroup is normal in its normalizer's normalizer chain; the new definition of $N(\eta)$ gives a clean base for iterated normalizers and perhaps for pronormality.
The paper focuses on L-subgroups of an L-group; the same conjugate construction could be applied to L-subsets of other L-algebras, such as L-ideals of L-rings, where normality and normalizer-like operators are defined, though that is not done here.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

Short version: this paper is a real extension of lattice-valued fuzzy subgroup theory, and its main normalizer identity is worth taking seriously. But the paper has a definitional inconsistency that has to be fixed before the work is usable.

What is actually new: the conjugate of an L-subgroup by an L-point of the parent L-group, rather than by an ordinary group element. That definition does not appear in [15,17], and it specializes the conjugate-by-L-subset from [11] in a way that preserves L-subgroups. Theorem 3.2 establishes the preservation, 3.4 and 3.5 give the expected homomorphism behavior, Proposition 4.1 gives the normality test, and Theorem 4.4—N(η)^{a_z}(g)=a∧N(η^{a_z})(g)—is the deepest result. I checked the long proof of 4.4; the steps are valid, and complete distributivity is used precisely where arbitrary meets and joins are interchanged.

The soft spots. First, the stress-test note lands. Definition 3.1 as written gives a∧η(z^{-1}xz) for the conjugate of η by a_z, while Theorem 3.2 and the rest of the paper use a∧η(zxz^{-1}). The paper claims the latter follows 'in view of Definition 3.1,' but it does not; it is the conjugate by a_{z^{-1}}. Since every L-point a_z in an L-subgroup has its inverse partner a_{z^{-1}} with the same strength, all theorems survive a global z↔z^{-1} change. So the math is not wrong, but the foundational definition and the notation are at odds. The authors must state their convention explicitly.

Second, Theorem 3.6 contains a genuinely miswritten inequality: η^{a_z}(x)≥a∧η(e) is asserted in the forward direction. That is generally false. The conclusion still follows from η^{a_z}(x)≥t directly, so the repair is local.

The reliance on earlier papers by the same group for normalizer, generated L-subgroups, and maximal L-subgroups is proper; those are established foundations, and the citations fit.

Who this is for: people working in L-fuzzy algebra, especially the Ajmal–Jahan line. The significance is modest because the area is small, but the contribution within that area is genuine.

Recommendation: send to peer review, with an explicit referee request to reconcile Definition 3.1 with the formula used in Theorem 3.2. It is the kind of flaw that a serious referee should catch and the authors can fix quickly.

Referee Report

3 major / 5 minor

Summary. The paper develops a notion of conjugation for L-subgroups of an L-group, where the conjugating object is an L-point a_z of the parent L-group rather than an ordinary group element. Section 3 introduces the definition, proves that the conjugate object is again an L-subgroup (Theorem 3.2), studies behavior under homomorphisms (Theorems 3.4 and 3.5), gives a level-set characterization (Theorem 3.6), and treats the conjugate of a maximal L-subgroup for a chain L (Theorem 3.9). Section 4 connects conjugacy to normality: Proposition 4.1 characterizes normal L-subgroups by containment of all conjugates, Lemma 4.8 relates commutativity of cosets to inclusion of conjugates, Theorem 4.4 proves the identity N(η)^{a_z}(g)=a∧N(η^{a_z})(g), and Definition 4.9 redefines the normalizer using conjugacy. The paper is written in the framework of Ajmal and Jahan's earlier L-group theory, with complete distributivity of L as a standing assumption.

Significance. If the results are correctly stated, the paper supplies a natural L-valued analogue of conjugate subgroups and shows that this notion interacts well with the already-established normalizer theory from [5]. The most valuable result is Theorem 4.4, whose compatibility identity N(η)^{a_z}=a∧N(η^{a_z}) is a substantive check that the new conjugation operation and the normalizer operation commute in the expected sense. The examples are concrete and helpful. However, the paper is not currently self-consistent: the displayed formula for the conjugate by an L-point does not follow from Definition 3.1 as written, and this discrepancy propagates through nearly every subsequent statement. The main ideas are defensible, but the text needs a systematic correction before the claims can be accepted as stated.

major comments (3)

[Definition 3.1 and the formula after it] There is a direct contradiction between Definition 3.1 and the formula used throughout. Definition 3.1 states θηθ^{-1}(x)=∨_{x=zyz^{-1}}{η(y)∧θ(z)}. For θ=a_z this evaluates to a∧η(z^{-1}xz), not a∧η(zxz^{-1}). The text immediately defines η^{a_z}(x)=a∧η(zxz^{-1}) and says this is 'in view of Definition 3.1', which is incorrect; that value is obtained from θ=a_{z^{-1}}, not θ=a_z. The subsequent Theorem 3.2, Theorems 3.4-3.6, Proposition 4.1, Lemma 4.8, Definition 4.9, and Theorem 4.4 all use the formula with zxz^{-1}. Since the central object of the paper is defined inconsistently with its own cited definition, this must be fixed globally. One option is to define η^{a_z} explicitly as the L-subset obtained from Definition 3.1 with the L-point a_{z^{-1}}; another is to adjust Definition 3.1 and then re-check the statements of Theorems 3.4 and 3.5, where f(z) and its inverse are currently conflated.
[Theorem 3.6, forward direction] The proof of (⇒) contains the assertion that for x∈ν_t, η^{a_z}(x)=a∧η(zxz^{-1})≥a∧η(e), which is not true for a general x≠e. The intended conclusion still follows directly: from ν(x)=a∧η(zxz^{-1})≥t and t≤a, the meet property gives η(zxz^{-1})≥t. The proof should be rewritten to use this argument instead of the false inequality. This matters because Theorem 3.6 is used later in Lemma 4.7 and Definition 4.9.
[Theorem 3.9] The proof that the constructed γ is an L-subgroup contains an invalid lattice inequality. The line γ(xy)≥{η(x)∧η(y)}∨{θ(z^{-1}xz)∧θ(z^{-1}yz)}≥{η(x)∨θ(z^{-1}xz)}∧{η(y)∨θ(z^{-1}yz)} is not valid in general lattices; for example, with p=1, r=0, q=0, s=1 in a two-element lattice, (p∧q)∨(r∧s)=0 while (p∨r)∧(q∨s)=1. Thus the argument does not establish that γ∈L(μ), and the contradiction to maximality is not obtained. This theorem is not used later in the paper, but it is stated and proved as a result and needs either a corrected proof or a corrected statement.

minor comments (5)

[Theorem 3.6] The notation η_t^{z^{-1}} is used without prior definition; the set-theoretic conjugate of a level subgroup should be defined explicitly before the theorem.
[Theorem 3.6 statement] The phrase 'for a_z∈μ' should be existential, e.g. 'for some a_z∈μ', to avoid the impression that a,z are arbitrary but fixed in the statement.
[Example 2] The sentence 'the normalizer of N(η^{a_r})' should presumably read 'the normalizer of η^{a_r}', since the displayed computation is for N(η^{a_r}).
[Conclusion] The final paragraph makes an unsupported broad historical claim that fuzzy group theory 'came to a halt' after Head's metatheorem and that the metatheorem and subdirect product theorem are not applicable in the L-setting; this should be removed or supported by precise references.
[Throughout] There are several typographical and grammatical slips ('arbitray', missing spaces around exponents, inconsistent use of 'L-fuzzy subgroups') that should be corrected in a final version.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central normalizer-conjugacy identity is proved from the cited normalizer definition, not assumed.

full rationale

The derivation chain is self-contained at the level of the target results. The conjugate η^{a_z} is introduced as a specialization of Definition 3.1 (from [11]), and Theorem 3.2 verifies from that definition that it is an L-subgroup of µ; the later results (Theorems 3.3–3.9, Proposition 4.1) manipulate this defined object directly. Theorem 4.4, the deepest claim, takes the normalizer N(η) exactly as defined in [5] (Definition 4.3) and proves N(η)^{a_z}(g)=a∧N(η^{a_z})(g) by comparing the two sets of L-points whose cosets commute; it does not assume the identity or define N(η^{a_z}) by it. Definition 4.9 restates the normalizer via conjugacy, but only after Lemma 4.8 proves equivalence with the coset-commutation normalizer of [5], so it is a characterization rather than a circular redefinition. The self-citations to [1], [5], and [11] provide prior definitions and a generation formula, but the main conjugacy-normalizer identity is not imported from those papers; it is derived here. A separate non-circularity concern: the displayed formula η^{a_z}(x)=a∧η(zxz^{-1}) does not literally follow from Definition 3.1 as written, which would yield a∧η(z^{-1}xz); this is an internal sign-convention inconsistency, but it does not make any result equivalent by construction to its own input, and it propagates uniformly through the paper.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted and no new primitive entities are postulated. The conjugate L-subgroup is a defined construction from existing objects: an L-subgroup, an L-point, and the lattice meet.

assumptions (4)

domain assumption L is a completely distributive lattice with extremes 1 and 0.
Stated in Section 2; used to interchange arbitrary joins with meets in Theorems 3.4 and 4.4.
standard math Standard definitions and level-subset characterizations of L-subgroups and normal L-subgroups are accepted from the literature.
Invoked throughout, for example to identify L-subgroups by their level subsets via Theorems 2.2, 2.5, 2.9, and 2.11.
domain assumption Theorem 3.9 additionally assumes L is a chain.
The proof needs a∧b to equal one of a or b; stated in the theorem.
domain assumption The normalizer N(η), defined in [5] as the union of L-points a_x with a_x∘η=η∘a_x, is the largest L-subgroup of µ in which η is normal.
Definition 4.3 is quoted from [5] and the new conjugate-based Definition 4.9 is shown equivalent via Lemma 4.8; Theorem 4.4 presupposes that characterization.

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Cite this review

Pith. "Pith review of Conjugate L-Subgroups of an L-group and their Applications to Normality and Normalizer." pith.science (2026). https://pith.science/paper/IIMCR42X

@misc{pith2026250620692,
  author       = {Pith},
  title        = {Pith review of: Conjugate L-Subgroups of an L-group and their Applications to Normality and Normalizer},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IIMCR42X}},
  note         = {Machine review of arXiv:2506.20692}
}

read the original abstract

In this paper, the notion of the conjugate of an L-subgroup by an L-point has been introduced. Then, several properties of conjugate L-subgroups have been studied analogous to their group-theoretic counterparts. Also, the notion of conjugacy has been investigated in the context of normality of L-subgroups. Furthermore, some important relationships between conjugate L-subgroups and normalizer have also been established. Finally, the normalizer of an L-subgroup of an L-group has been defined by using the notion of conjugate L-subgroups.

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Works this paper leans on

19 extracted references · 19 canonical work pages · cited by 1 Pith paper

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [11]

Davvaz, N

I, Jahan, B. Davvaz, N. Ajmal,NilpotentL-subgroups Satisfy the Normalizer Condition, Journal of Intelligent and Fuzzy Systems, 33 (2017), 1841–1854

work page 2017

[2] [5]

Ajmal, I

N. Ajmal, I. Jahan,AnL-point characterization of normality and normalizer of anL- subgroup of anL-group,Fuzzy Information and Engineering, 6 (2014), 147-166

work page 2014

[3] [1]

Ajmal, I

N. Ajmal, I. Jahan,GeneratedL-subgroup of anL-group,Iranian Journal of Fuzzy Systems, 12(2) (2015), 129-136

work page 2015

[4] [2]

Ajmal, I

N. Ajmal, I. Jahan,A study of normal fuzzy subgroups and characteristic fuzzy subgroups of a fuzzy group,Fuzzy Information and Engineering, 2 (2012) 123–143

work page 2012

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Ajmal, I

N. Ajmal, I. Jahan,Normal Closure of anL-subgroup of anL-group,The Journal of Fuzzy Mathematics, 22 (2014), 115-126

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Ajmal, I

N. Ajmal, I. Jahan,Nilpotency and theory ofL-Subgroups of anL-group,Fuzzy Information and Engineering, 6 (2014), 1-17

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