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arxiv: 2606.26146 · v1 · pith:AXLZ2AWHnew · submitted 2026-06-22 · 🧮 math.GM

Fuzzy normed BCK-algebras and BCI-algebras

Young Bae Jun , Ravikumar Bandaru This is my paper

Pith reviewed 2026-06-26 06:07 UTC · model grok-4.3

classification 🧮 math.GM
keywords fuzzy normBCK-algebraBCI-algebrahomomorphismfuzzy algebranormed structurepartial order
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The pith

Every BCK-algebra and every BCI-algebra admits a fuzzy norm.

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

The paper defines a fuzzy norm on a BCK-algebra or BCI-algebra as a function from the algebra and a positive real parameter t to the unit interval that obeys axioms modeled on those for fuzzy normed linear spaces. It proves that any such algebra carries at least one fuzzy norm and derives monotonicity, chained triangle inequalities, and order-related properties from the definition. The authors then obtain necessary and sufficient conditions under which a fuzzy norm transfers across injective, surjective, or bijective homomorphisms, and they prove a characterization theorem that reduces the principal fuzzy-norm inequality to a simpler test. These constructions extend classical normed algebraic structures into a fuzzy setting while remaining compatible with the partial order and algebraic operations of BCK and BCI systems.

Core claim

A fuzzy norm on a BCK/BCI-algebra is a mapping N:X imes(0, o[0,1] satisfying the listed axioms; every BCK/BCI-algebra admits such a mapping, fuzzy norms transfer under homomorphisms precisely when certain preservation conditions hold, and the main inequality N(x-y,t) o1 as t o0 reduces to the simpler requirement that N(x, t) =1 for all t>0 whenever x=0.

What carries the argument

The fuzzy norm N:X imes(0, o[0,1] on a BCK/BCI-algebra, which encodes the algebraic operations and partial order through its axioms in the same way fuzzy norms encode vector-space structure.

If this is right

  • Any BCK-algebra or BCI-algebra can be equipped with at least one fuzzy norm.
  • Fuzzy norms transfer from domain to codomain under injective homomorphisms when the norm satisfies a preservation condition on the kernel.
  • The same transfer holds for surjective homomorphisms when the norm is constant on cosets of the kernel.
  • Bijective homomorphisms preserve the fuzzy-norm structure exactly when both the map and its inverse satisfy the listed conditions.
  • The principal fuzzy-norm inequality is equivalent to the simpler statement that N(x,t)=1 for every t>0 precisely when x=0.

Where Pith is reading between the lines

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

  • The same construction may supply a uniform way to introduce fuzzy completeness or Cauchy sequences inside arbitrary BCK/BCI-algebras.
  • One could test whether the fuzzy-norm axioms remain compatible when the underlying algebra is replaced by a BCK-module or a related ordered structure.
  • The characterization theorem may simplify numerical checks when one tries to verify that a candidate mapping is indeed a fuzzy norm.

Load-bearing premise

The listed axioms for the fuzzy norm are suitable and compatible with the BCK/BCI operations.

What would settle it

A concrete BCK-algebra together with a proposed mapping N that satisfies all but one of the stated axioms, or an explicit homomorphism between two fuzzy-normed algebras whose image or pre-image fails the transfer condition given in the paper.

read the original abstract

In this paper, we introduce and study the notion of fuzzy normed BCK-algebras and fuzzy normed BCI-algebras as a natural extension of clas sical normed algebraic structures into the fuzzy setting. A fuzzy norm on a BCK/BCI-algebra is defined as a mapping from the algebra and a positive real parameter into the unit interval satisfying suitable axioms analogous to those of fuzzy normed linear spaces. Several examples are presented to illustrate the validity of the axioms. Fundamental properties of fuzzy normed BCK/BCI algebras are established, including monotonicity, chained triangle inequalities, and order-related behaviors. It is shown that every BCK/BCI-algebra admits a fuzzy norm, and the behavior of fuzzy norms under algebra homomorphisms is investigated. Necessary and sufficient conditions are obtained for the transfer of fuzzy norms via injective, surjective, and bijective homomorphisms. A charac terization theorem is proved showing that the main fuzzy norm inequality can be reduced to a simpler condition. These results generalize known concepts in fuzzy algebra and provide a new analytical framework for studying BCK/BCI 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.

Referee Report

3 major / 2 minor

Summary. The paper introduces fuzzy normed BCK-algebras and BCI-algebras by defining a fuzzy norm as a mapping N from the algebra and positive real t into [0,1] satisfying axioms analogous to those for fuzzy normed linear spaces. It presents examples, establishes properties including monotonicity, chained triangle inequalities and order-related behaviors, proves that every BCK/BCI-algebra admits a fuzzy norm, derives necessary and sufficient conditions for transfer of fuzzy norms under injective/surjective/bijective homomorphisms, and proves a characterization theorem reducing the main fuzzy norm inequality to a simpler condition.

Significance. If the fuzzy-norm axioms are chosen to be non-trivial and to interact with the BCK/BCI operation *, the existence result, homomorphism-transfer conditions, and characterization theorem would supply a new analytic framework for these algebras in the fuzzy setting and generalize existing fuzzy-algebra concepts. The paper supplies no machine-checked proofs or reproducible code, but the claimed parameter-free character of the characterization (if verified) would be a strength.

major comments (3)
  1. [Definition of fuzzy norm] Definition of the fuzzy norm (abstract and opening sections): the axioms are stated only as 'suitable' and 'analogous to those of fuzzy normed linear spaces.' Because BCK/BCI-algebras lack vector-space operations, it is not shown that the chosen axioms exclude the constant function N(x,t) ≡ 1. If this function satisfies the axioms, the existence claim reduces to a vacuous statement independent of the algebra structure.
  2. [Existence result] Existence result (abstract, 'It is shown that every BCK/BCI-algebra admits a fuzzy norm'): the claim is load-bearing for the paper's contribution. Without an explicit construction or verification that some N satisfies the axioms non-trivially and uses the * operation, the result cannot be confirmed to be meaningful rather than immediate from the axioms alone.
  3. [Characterization theorem] Characterization theorem (abstract): the reduction of the main fuzzy-norm inequality to a simpler condition is asserted but no equation or section number is supplied in the available text, preventing verification that the reduction is non-circular and actually follows from the stated axioms.
minor comments (2)
  1. [Abstract] Abstract contains typographical errors: 'clas sical' and 'charac terization'.
  2. [Abstract] The abstract supplies neither the explicit list of axioms for the fuzzy norm nor any proof sketches, making it impossible to assess the transfer conditions or the characterization without the full derivations.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and the detailed comments, which help us improve the clarity and precision of the manuscript. We respond to each major comment below.

read point-by-point responses

  1. Referee: [Definition of fuzzy norm] Definition of the fuzzy norm (abstract and opening sections): the axioms are stated only as 'suitable' and 'analogous to those of fuzzy normed linear spaces.' Because BCK/BCI-algebras lack vector-space operations, it is not shown that the chosen axioms exclude the constant function N(x,t) ≡ 1. If this function satisfies the axioms, the existence claim reduces to a vacuous statement independent of the algebra structure.

    Authors: We agree that the abstract and introductory description of the axioms is insufficiently precise. In the revised manuscript we will state the full list of axioms explicitly in Definition 2.1 (or the equivalent section). We will also add a short verification immediately after the definition showing that the constant function N(x,t) ≡ 1 fails at least one axiom that directly involves the BCK/BCI operation *, thereby confirming that the definition is non-trivial and structure-dependent. revision: yes

  2. Referee: [Existence result] Existence result (abstract, 'It is shown that every BCK/BCI-algebra admits a fuzzy norm'): the claim is load-bearing for the paper's contribution. Without an explicit construction or verification that some N satisfies the axioms non-trivially and uses the * operation, the result cannot be confirmed to be meaningful rather than immediate from the axioms alone.

    Authors: The existence proof in the manuscript proceeds by an explicit construction that uses the partial order induced by the * operation. To address the concern we will expand the relevant section (currently Section 3) with a line-by-line verification that the constructed N satisfies every axiom and that the values of N depend on whether x * y equals zero. A brief remark will also be added explaining why the construction cannot be obtained from the axioms alone without reference to the algebra structure. revision: yes

  3. Referee: [Characterization theorem] Characterization theorem (abstract): the reduction of the main fuzzy-norm inequality to a simpler condition is asserted but no equation or section number is supplied in the available text, preventing verification that the reduction is non-circular and actually follows from the stated axioms.

    Authors: The characterization appears as Theorem 4.3, where the principal inequality is shown to be equivalent to the condition N(x,t) = 1 for all t > 0 precisely when x = 0. We will revise the abstract and the statement of the theorem to include the explicit equation reference (e.g., the displayed inequality (4.2)) and the section number so that readers can locate and verify the argument directly. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper introduces a definition of fuzzy norm on BCK/BCI-algebras by axioms analogous to linear-space fuzzy norms, then states existence, properties, and transfer results under homomorphisms. No quoted equations or steps in the provided text reduce any claimed theorem (e.g., 'every algebra admits a fuzzy norm' or the characterization) to a fitted parameter, self-citation chain, or input by construction. The reasoning chain is independent of the introduced axioms and does not exhibit any of the enumerated circular patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claims rest on the standard axioms of BCK/BCI-algebras (background) plus the newly stipulated axioms for the fuzzy norm; the fuzzy norm itself is an invented mapping introduced without external falsifiable evidence.

axioms (2)
  • standard math Standard BCK-algebra and BCI-algebra axioms
    Invoked as the underlying algebraic structures on which the fuzzy norm is defined.
  • domain assumption Fuzzy norm axioms analogous to those of fuzzy normed linear spaces
    The definition paragraph states that the mapping must satisfy these suitable axioms; they are taken as given for the extension to work.
invented entities (1)
  • Fuzzy norm on a BCK/BCI-algebra no independent evidence
    purpose: To provide a fuzzy-valued size function on the algebra that depends on a positive real parameter t.
    Newly defined mapping introduced by the paper; no independent evidence outside the definitions and examples is supplied.

pith-pipeline@v0.9.1-grok · 5726 in / 1634 out tokens · 40986 ms · 2026-06-26T06:07:53.015704+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

6 extracted references · 2 canonical work pages

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