Pith. sign in

REVIEW 3 major objections 3 minor 45 references

Uninorms via two comparable closure operators on bounded lattices

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

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read With two comparable closure operators, a piecewise operation on a bounded lattice is a uninorm exactly when both operators keep their outputs out of [e,1].

desk verdict The main two-operator construction is new and Theorem 3.1's proof mostly holds, but the paper's showcase example is invalid and the dual theorems are unproved. read the letter →

arxiv 2505.04220 v1 pith:2ENP6LHW submitted 2025-05-07 math.FA math.LO

Zhenyu Xiu , Xu Zheng This is my paper · ORCID
classification math.FAmath.LO MSC 06B0503E72
keywords boundedlatticesuninormsclosureoperatorsinteriort-conormst-normsneutralelementpiecewiseconstructions
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

The paper establishes necessary-and-sufficient recipes for building uninorms on bounded lattices from two comparable closure operators, and dually from two comparable interior operators. The central construction fixes a neutral element $e$ and a t-conorm on $[e,1]$, then assigns mixed-region pairs $]0,e[\times]e,1]$ and $I_e\times]e,1]$ to the two closure operators rather than to one of the inputs. Theorem 3.1 identifies exactly when this piecewise operation is associative and monotone: $cl_1(x)\notin[e,1]$ for every $x\in]0,e[$, and $cl_2(x)\notin[e,1]$ for every $x\in I_e$. The interest is that the resulting uninorms need not belong to the standard classes $U_{\min}^{*}\cup U_{\min}^{1}$ (or their duals), unlike every earlier closure-operator construction, and the identity case of the recipe reproduces the classical constructions.

What carries the argument

The load-bearing object is the piecewise operation $U$ built from a t-conorm on $[e,1]$ and two closure operators assigned to the two lower blocks of the lattice, namely $]0,e[$ and the set $I_e$ of elements incomparable with $e$. Closure operators are the monotone, inflationary, idempotent maps; their defining identity $cl(cl(x)\wedge y)=cl(x)$ whenever $x\le y$ is what makes the crossed terms $cl_i(x)\wedge(x\vee e)$ associate correctly. The comparability condition $cl_1\le cl_2$ on $L\setminus[e,1]$ orders the block outputs, and the exclusion conditions $cl_1(x)\notin[e,1]$ and $cl_2(x)\notin[e,1]$ force those outputs to stay inside the assigned blocks $]0,e[$ and $I_e$, respectively. Together these ingredients reduce a seemingly arbitrary binary operation to a four-block structure whose associativity can be checked block by block.

What would settle it

On a candidate finite lattice, compute $U$ as in Theorem 3.1 and test the two exclusion conditions. If some $x\in]0,e[$ has $cl_1(x)\in[e,1]$, then $U(U(1,x),x)=U(cl_1(x)\wedge(x\vee e),x)=x$ while $U(1,U(x,x))=U(1,0)=0$, so associativity fails; the analogous computation with $cl_2$ on $I_e$ refutes the dual condition. A single concrete lattice table exhibiting this failure would settle the claim by showing the conditions are truly necessary.

Watch

Extended reading notes

Core claim

For $e\in L\setminus\{0,1\}$, $S$ a t-conorm on $[e,1]$, and $cl_1\le cl_2$ pointwise on $L\setminus[e,1]$, the operation defined by $S$ on $[e,1]^2$, by $x$ or $y$ on the strips meeting $\{e\}$, by $cl_1(x)\wedge(x\vee e)$ on $]0,e[\times]e,1]$ and its transpose, by $cl_2(x)\wedge(x\vee e)$ on $I_e\times]e,1]$ and its transpose, and by $0$ elsewhere, is a uninorm with neutral element $e$ if and only if the two exclusion conditions hold. The necessity is detected by associativity at the triple $(1,x,x)$: if $cl_1(x)\in[e,1]$, then $U(U(1,x),x)=x$ whereas $U(1,U(x,x))=0$. The sufficient direction splits $L$ into the regions $]0,e[$, $\{e\}$, $I_e$, and $]e,1]$, checks increasingness case by case using $cl_1\le cl_2$, and proves associativity with the closure identity $cl(cl(x)\wedge y)=cl(x)$ for $x\le y$. The dual interior-operator theorems mirror the construction on $[0,e]$ with a t-norm, meet replaced by join, and $1$ as the absorbing value.

Load-bearing premise

The argument leans on the premise that $cl_1(x)\le cl_2(x)$ for every element outside $[e,1]$; the monotonicity proof moves inequalities from one block to the next through this chain, and without that comparability the necessary-and-sufficient characterization is not established.

Editorial extensions

If this is right

  • When both closure operators are the identity map, the construction in Theorem 3.1 reduces exactly to the standard uninorm $U_s$ built from $S$ on $[e,1]$ and $0$ elsewhere.
  • Setting only $cl_1$ to the identity yields a single-closure-operator uninorm that lies in $U_{\min}^{*}$ but not in $U_{\min}\cup U_{\max}^{r}$, as Example 3.2 demonstrates.
  • For lattices whose interval $]0,e[$ is a singleton or is totally incomparable internally, the constructed uninorm belongs to $U_{\min}^{*}$, and the analogous structural condition places the Theorem 3.3 construction into $U_{\min}^{1}$.
  • When the top element is made absorbing in Theorem 3.3, the extra condition $S(x,y)<1$ for all $x,y\in]e,1[$ is forced, and it is exactly as necessary as the closure-operator exclusion conditions.
  • The interior-operator mirrors, Theorems 3.2 and 3.4, produce the corresponding dual facts with a t-norm $T$ on $[0,e]$ and with $1$ or $0$ as the absorbing element.

Reading between the lines

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

  • Beyond the paper: the same block-decomposition proof would work for any pair of increasing, inflationary, idempotent endomaps assigned to the two regions, so the result is likely a special case of a general two-monotone-block uninorm recipe.
  • Beyond the paper: because the theorem is an if-and-only-if statement, a finite-lattice test needs only to check comparability and the two exclusion conditions, not all associativity triples, to certify that a proposed piecewise operation is a uninorm.
  • Beyond the paper: replacing the top element $1$ by an arbitrary fixed element in Theorem 3.3 suggests a family of absorbing-point uninorms whose characterization would combine a condition on the t-conorm with the closure-operator conditions.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 3 minor

Summary. The paper proposes constructions of uninorms on bounded lattices from pairs of comparable closure operators (Theorems 3.1 and 3.3) and, dually, pairs of comparable interior operators (Theorems 3.2 and 3.4), together with a t-conorm on [e,1] or a t-norm on [0,e]. The main result, Theorem 3.1, defines a piecewise operation U and characterizes, under the standing comparability assumption cl1(x) ≤ cl2(x) for x ∈ L \ [e,1], exactly when U is a uninorm with neutral element e: namely, cl1(x) ∉ [e,1] for all x ∈ ]0,e[ and cl2(x) ∉ [e,1] for all x ∈ Ie. Several degenerate cases with a single operator are derived, and the paper claims that the resulting uninorms need not lie in the previously studied classes U_min* ∪ U_min^1 or U_max* ∪ U_max^0. Examples and tables are provided to illustrate the constructions.

Significance. If the main theorems are correct, the construction is a genuine addition to the known toolkit: in contrast to earlier closure- and interior-operator based uninorms, the present values on the mixed region ]0,e[ × ]e,1] actually depend on the chosen operators, and the conditions on those operators are necessary and sufficient. The proof of Theorem 3.1 is largely detailed: the necessity arguments are clean, the associativity proof is organized by Proposition 2.3, and Lemma 3.1 is used appropriately. The paper also includes a useful consistency check: setting the operators to the identity recovers the Karacal-Mesiar uninorm. However, the significance is substantially tempered by the fact that the only displayed non-degenerate example is invalid, and the dual interior-operator theorems are stated without proof. The central theorem may well be correct, but the manuscript in its present form does not provide a reliable non-degenerate demonstration of the construction.

major comments (3)
  1. [Example 3.1 (Tables 1 and 2)] The closure operator cl2 in Table 1 is not a closure operator. Since a ∈ ]0,e[ and j ∈ ]e,1], axiom (CL2) applied to a and e gives cl2(a∨e) = cl2(e) = e, while cl2(a)∨cl2(e) = j∨e = j, forcing e = j, a contradiction; the same failure occurs for b. Moreover, Table 2 is not the operation defined in Theorem 3.1 for the stated operators: Table 2 has U(k,k) = k, which forces k ∈ [e,1], and U(m,m) = 0, so m ∈ ]0,e[ ∪ Ie; but for m ∈ ]0,e[ the theorem gives U(m,k) = cl1(m)∧e = k∧e = e, and for m ∈ Ie it gives U(m,k) = cl2(m)∧(m∨e) = k∧(m∨e) = m∨e ≠ 0, whereas Table 2 lists U(m,k) = 0. The example is therefore not a valid instance of Theorem 3.1 and cannot support the Conclusion's claim that nontrivial closure operators satisfying the hypotheses exist; it should be corrected or replaced.
  2. [Theorems 3.2 and 3.4] The dual interior-operator theorems are stated without proof. The abstract and the body advertise these results as part of the contribution, and the claimed characterizations (int2(x) ∉ [0,e] for x ∈ Ie and int1(x) ∉ [0,e] for x ∈ ]e,1[, together with 0 < T(x,y) in Theorem 3.4) are not verified by any argument. Calling these results 'dual' is not a substitute for a proof, because the constructions use different intervals and the necessity arguments for the interior-operator conditions require a separate check. Please supply the proofs or a precise order-reversal derivation from Theorems 3.1 and 3.3.
  3. [Theorem 3.3 proof] The sufficiency direction of Theorem 3.3(1) is only a sketch: it says that, taking Theorem 3.1 into account, it is enough to check the cases with 1 ∈ {x,y,z}, and then asserts monotonicity and associativity for those cases without demonstration. Since Theorem 3.3 differs from Theorem 3.1 by using [e,1[ in place of [e,1] for the t-conorm region and by making 1 absorbing, the extra condition S(x,y) < 1 is not merely cosmetic; its role in the sufficiency proof should be shown explicitly, including the mixed cases involving 1. Please expand this part of the proof or state clearly that it is a direct analogue with the same case analysis.
minor comments (3)
  1. [Theorem 3.1, monotonicity case 1.2.3] The inequality U(x,z) ≤ U(y,z) in case 1.2.3 is asserted without explanation; the reader should be shown the chain cl1(x)∧e ≤ cl2(y)∧e ≤ cl2(y)∧(y∨e), which uses the standing comparability cl1(x) ≤ cl2(x) and monotonicity of cl2. This is the one place where the comparability assumption is genuinely load-bearing.
  2. [Proposition 3.1] The phrase ']0,e[ ⊆ {x} for some x ∈ L' is awkward; it should be phrased as ']0,e[ is a singleton' or ']0,e[ = {x}'. The same applies in Propositions 3.3, 3.6, and 3.9.
  3. [Example 3.3 (Tables 4 and 5)] A similar consistency check to the one for Example 3.1 should be performed for Example 3.3. In particular, Table 5 has U(c,c) = c, which suggests c ∈ ]e,1[, but then cl2(l) = c and cl2(m) = c for l,m that appear to lie in Ie would violate the hypothesis cl2(x) ∉ [e,1] required by Theorem 3.3; the example should be recomputed or clarified.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main characterization is proved from definitions, not derived from its own conclusion.

full rationale

The paper's central result, Theorem 3.1, defines a candidate operation U from a t-conorm S and two closure operators cl1, cl2 and proves an if-and-only-if characterization of when U is a uninorm with neutral element e. The necessity direction derives the conditions cl1(x) not in [e,1] for x in ]0,e[ and cl2(x) not in [e,1] for x in Ie from explicit associativity contradictions, e.g. U(U(1,x),x) = U(e,x) = x versus U(1,U(x,x)) = U(1,0) = 0. The sufficiency direction verifies monotonicity and associativity directly, using Lemma 3.1 and Proposition 2.3; the comparability assumption cl1 <= cl2 is a stated hypothesis used inside the proof, not a conclusion imported from outside. No parameter is fitted to the desired result and no quantity is 'predicted' from data that already determines it. The degenerate cases with cl1 = cl2 or identity closures reduce to the known Karaçal–Mesiar construction; this is explicitly described as a consistency check and does not bootstrap the main theorem. The authors' own prior works are cited only for context and not as load-bearing evidence for Theorem 3.1. Even the illustrative defect noted in Example 3.1's Table 1, where cl2 may fail axiom CL2, would be a correctness issue for an example, not a circular derivation. Consequently, no circular step can be identified in the paper's derivation chain.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters are introduced; the construction quantifies over arbitrary closure/interior operators and t-norms/t-conorms. The only inputs are the given lattice, neutral element, and operators, all of which are standard objects in the literature. No new entities are postulated.

assumptions (6)
  • standard math Classical set-theoretic mathematics with standard bounded lattice theory (order, meet, join, intervals).
    Used throughout; e.g., Section 2 Definitions 2.1 and 2.2 define bounded lattices and intervals.
  • standard math Closure operators satisfy (CL1) x≤cl(x), (CL2) cl(x∨y)=cl(x)∨cl(y), (CL3) cl(cl(x))=cl(x) (Everett [20]).
    This definition underlies Lemma 3.1 and all construction theorems.
  • standard math Interior operators satisfy (IN1) int(x)≤x, (IN2) int(x∧y)=int(x)∧int(y), (IN3) int(int(x))=int(x).
    Used in the dual Theorem 3.2 and Proposition 3.4.
  • standard math Proposition 2.3 from Ji [28]: a commutative operation on a union of subsets is associative iff associativity holds on all local triples and mixed pairs.
    Load-bearing in the associativity proofs of Theorems 3.1 and 3.3, which reduce the verification to a finite list of cases.
  • domain assumption A t-conorm S on [e,1] with neutral element e is given as input in Theorems 3.1 and 3.3.
    The construction assumes such an S exists; it is not derived. Similarly, a t-norm T on [0,e] is assumed in the dual results.
  • domain assumption The two closure operators are comparable on L\[e,1]: cl1(x) ≤ cl2(x) for all x∈L\[e,1].
    This comparability is a standing premise of Theorems 3.1 and 3.3 and is used in the monotonicity proof, e.g. case 1.2.3.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Uninorms via two comparable closure operators on bounded lattices." pith.science (2026). https://pith.science/paper/2ENP6LHW

@misc{pith2026250504220,
  author       = {Pith},
  title        = {Pith review of: Uninorms via two comparable closure operators on bounded lattices},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2ENP6LHW}},
  note         = {Machine review of arXiv:2505.04220}
}
abstract

In this paper, we propose novel methods for constructing uninorms using two comparable closure operators or, alternatively, two comparable interior operators on bounded lattices. These methods are developed under the necessary and sufficient conditions imposed on these operators. Specifically, the construction of uninorms for $(x ,y )\in ]0 ,e [\times]e ,1 [ \cup ]e ,1 [\times]0 ,e [$ depends not only on the structure of the bounded lattices but also on the chosen closure operators (or interior operators). Consequently, the resulting uninorms do not necessarily belong to $\mathcal{U}_{min}^{*}\cup \mathcal{U}_{min}^{1}$ (or $\mathcal{U}_{max}^{*}\cup\mathcal{U}_{max}^{0}$). Moreover, we present the degenerate cases of the aforementioned results, which are constructed using only a single closure operator or a single interior operator. Some of these cases correspond to well-known results documented in the literature.

Figures

Figures reproduced from arXiv: 2505.04220 by the authors.

Figure 1
Figure 1. Fig.1., where [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 1
Figure 1. Fig.1. The uninorm [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Fig.2. The bounded lattice [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figures from the paper (5 more)
Figure 3
Figure 3. Figure 3: Fig.3. The bounded lattice [PITH_FULL_IMAGE:figures/full_fig_p011_3.png]
Figure 4
Figure 4. Figure 4: Fig.4. The uninorm [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]
Figure 5
Figure 5. Figure 5: Fig.5. The uninorm [PITH_FULL_IMAGE:figures/full_fig_p015_5.png]
Figure 6
Figure 6. Figure 6: Fig.6. The bounded lattice [PITH_FULL_IMAGE:figures/full_fig_p016_6.png]
Figure 7
Figure 7. Figure 7: Fig.7. The uninorm [PITH_FULL_IMAGE:figures/full_fig_p018_7.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

45 extracted references · 45 canonical work pages

  1. [1]

    A¸ sıcı, R

    E. A¸ sıcı, R. Mesiar, On the construction of uninorms on bounded lattices, Fuzzy Sets Syst. 408(2021)65–85

    work page 2021

  2. [2]

    Birkhoff, Lattice theory, 3rd Edition, Amer

    G. Birkhoff, Lattice theory, 3rd Edition, Amer. Math. Soc., Rhod e Island, 1967

    work page 1967

  3. [3]

    Bodjanova, M

    S. Bodjanova, M. Kalina, Construction of uninorms on bounded la ttices, in: IEEE 12th International Symposium on Intelligent Systems and Informa tics, SISY 2014, September 11–13, Subotica, Serbia, 2014, 61–66

    work page 2014

  4. [4]

    C ¸ aylı, A characterization of uninorms on bounded lattices by means of trian- gular norms and triangular conorms, Int

    G.D. C ¸ aylı, A characterization of uninorms on bounded lattices by means of trian- gular norms and triangular conorms, Int. J. Gen. Syst. 47(2018) 772–793

    work page 2018

  5. [5]

    C ¸ aylı, On the structure of uninorms on bounded lattices, Fu zzy Sets Syst

    G.D. C ¸ aylı, On the structure of uninorms on bounded lattices, Fu zzy Sets Syst. 357(2019)2–26

    work page 2019

  6. [6]

    C ¸ aylı, Alternative approaches for generating uninorms on bounded lattices, Inf

    G.D. C ¸ aylı, Alternative approaches for generating uninorms on bounded lattices, Inf. Sci. 488(2019)111–139

    work page 2019

  7. [7]

    C ¸ aylı, New methods to construct uninorms on bounded lattices, Int

    G.D. C ¸ aylı, New methods to construct uninorms on bounded lattices, Int. J. Approx. Reason. 115(2019)254–264

    work page 2019

  8. [8]

    C ¸ aylı, Uninorms on bounded lattices with the underlyingt-norms and t-conorms, Fuzzy Sets Syst

    G.D. C ¸ aylı, Uninorms on bounded lattices with the underlyingt-norms and t-conorms, Fuzzy Sets Syst. 395(2020)107–129

    work page 2020

Show all 45 references
  1. [9]

    C ¸ aylı, New construction approaches of uninorms on bounde d lattices, Int

    G.D. C ¸ aylı, New construction approaches of uninorms on bounde d lattices, Int. J. Gen. Syst. 50(2021)139–158. 20

    2021

  2. [10]

    C ¸ aylı, An alternative construction of uninorms on boundedlattices, Int

    G.D. C ¸ aylı, An alternative construction of uninorms on boundedlattices, Int. J. Gen. Syst. 52(5)(2023)574–596

    2023

  3. [11]

    C ¸ aylı, A characterization of uninorms on bounded lattices v ia closure and in- terior operators, Kybernetika 59(5)(2023)768–790

    G.D. C ¸ aylı, A characterization of uninorms on bounded lattices v ia closure and in- terior operators, Kybernetika 59(5)(2023)768–790

    2023

  4. [12]

    C ¸ aylı, Constructing Uninorms on Bounded Lattices Through Closure and Inte- rior Operators, Int

    G.D. C ¸ aylı, Constructing Uninorms on Bounded Lattices Through Closure and Inte- rior Operators, Int. J. Uncertain. Fuzziness Knowl. Based Syst. 32(1)(2024)109-129

    2024

  5. [13]

    C ¸ aylı,¨U

    G.D. C ¸ aylı,¨U. Ertu˘ grul, F. Kara¸ cal, Some further construction methods for uninorms on bounded lattices, Int. J. Gen. Syst. 52(4)(2023)414–442

    2023

  6. [14]

    C ¸ aylı, F

    G.D. C ¸ aylı, F. Kara¸ cal, Construction of uninorms on bounded lattices, Kybernetika 53(2017)394–417

    2017

  7. [15]

    C ¸ aylı, F

    G.D. C ¸ aylı, F. Kara¸ cal, R. Mesiar, On a new class of uninorms on bounded lattices, Inf. Sci. 367(2016)221–231

    2016

  8. [16]

    Dan, B.Q

    Y.X. Dan, B.Q. Hu, A new structure for uninorms on bounded latt ices, Fuzzy Sets Syst. 386(2020)77–94

    2020

  9. [17]

    Dan, B.Q

    Y.X. Dan, B.Q. Hu, J.S. Qiao, New constructions of uninorms on bo unded lattices, Int. J. Approx. Reason. 110(2019)185–209

    2019

  10. [18]

    De Baets, J

    B. De Baets, J. Fodor, Van Melle’s combining function in MYCIN is a r epresentable uninorm: an alternative proof, Fuzzy Sets Syst. 104(1999)133– 136

    1999

  11. [19]

    Engelking

    R. Engelking. General Topology. Heldermann Verlag, Berlin, 198 9

  12. [20]

    Everett, Closure operators and Galois theory in lattices, T rans

    C.J. Everett, Closure operators and Galois theory in lattices, T rans. Am. Math. Soc. 55(1944)514–525

    1944

  13. [21]

    Grabisch, J.L

    M. Grabisch, J.L. Marichal, R. Mesiar, E. Pap, Aggregation func tions, Cambridge University Press, 2009

    2009

  14. [22]

    Grabisch, J.L

    M. Grabisch, J.L. Marichal, R. Mesiar, E. Pap, Aggregation func tions: construction methods, conjunctive, disjunctive and mixed classes, Inf. Sci. 18 1(2011)23–43

    2011

  15. [23]

    H´ ajek, Combining Functions for Certainty Degrees in Consu lting Systems, Int

    P. H´ ajek, Combining Functions for Certainty Degrees in Consu lting Systems, Int. J. Man-Mach. Stud. 22(1)(1985)59–76

    1985

  16. [24]

    H´ ajek, T

    P. H´ ajek, T. Havr´ anek, R. Jirou˘ sek, Uncertain Information Processing in Expert Systems, Boca Raton: CRC Press, 1992

    1992

  17. [25]

    P. He, X.P. Wang, Constructing uninorms on bounded lattices by using additive generators, Int. J. Approx. Reason. 136(2021)1–13

    2021

  18. [26]

    X.J. Hua, W. Ji, Uninorms on bounded lattices constructed by t- norms and t- subconorms, Fuzzy Sets Syst. 427(2022)109–131. 21

    2022

  19. [27]

    Construction of unin orms on bounded lattices

    X.J. Hua, H.P. Zhang, Y. Ouyang, Note on “Construction of unin orms on bounded lattices”, Kybernetika 57(2)(2021)372–382

    2021

  20. [28]

    Ji, Constructions of uninorms on bounded lattices by means o f t-subnorms and t-subconorms, Fuzzy Sets Syst

    W. Ji, Constructions of uninorms on bounded lattices by means o f t-subnorms and t-subconorms, Fuzzy Sets Syst. 403(2021)38–55

    2021

  21. [29]

    Kara¸ cal, ¨U

    F. Kara¸ cal, ¨U. Ertu˘ grul, R. Mesiar, Characterization of uninorms on bounded lat- tices, Fuzzy Sets Syst. 308(2017)54–71

    2017

  22. [30]

    Kara¸ cal, R

    F. Kara¸ cal, R. Mesiar, Uninorms on bounded lattices, Fuzzy Se ts Syst. 261(2015)33– 43

    2015

  23. [31]

    Klement, R

    E.P. Klement, R. Mesiar, E. Pap, A Universal Integral As Commo n Frame for Cho- quet and Sugeno Integral, IEEE Trans. Fuzzy Syst. 18(2010)17 8–187

    2010

  24. [32]

    Menger, Statistical metrics, Proc

    K. Menger, Statistical metrics, Proc. Natl. Acad. Sci. USA 8(1 942)535–537

  25. [33]

    Metcalfe, F

    G. Metcalfe, F. Montagna, Substructural Fuzzy Logics, J. S ymbolic Logic. 72(3)(2007)834–864

    2007

  26. [34]

    Ouyang, H.P

    Y. Ouyang, H.P. Zhang, Constructing uninorms via closure oper ators on a bounded lattice, Fuzzy Sets Syst. 395(2020)93–106

    2020

  27. [35]

    Pedrycz, K

    W. Pedrycz, K. Hirota, Uninorm-based logic neurons as adaptiv e and interpretable processing constructs, Soft Comput. 11(1)(2007)41–52

    2007

  28. [36]

    Saminger, On ordinal sums of triangular norms on bounded lat tices, Fuzzy Sets Syst

    S. Saminger, On ordinal sums of triangular norms on bounded lat tices, Fuzzy Sets Syst. 157(2006)1403–1416

    2006

  29. [37]

    Tak´ acs, Uninorm-Based Models for FLC Systems, J

    M. Tak´ acs, Uninorm-Based Models for FLC Systems, J. Intell. Fuzzy Syst. 19(1)(2008)65–73

    2008

  30. [38]

    Xie, S.J

    A.F. Xie, S.J. Li, On constructing the largest and smallest uninorm s on bounded lattices, Fuzzy Sets Syst. 386(2020)95–104

    2020

  31. [39]

    Xiu, Y.X

    Z.Y. Xiu, Y.X. Jiang, New structures for uninorms on bounded lat tices, J. Intell. Fuzzy Syst. 45(2)(2023)2019–2030

    2023

  32. [40]

    Z.Y. Xiu, X. Zheng, New construction methods of uninorms on bo unded lattices via uninorms, Fuzzy Sets Syst. 465(2023)108535

    2023

  33. [41]

    Z.Y. Xiu, X. Zheng, A new approach to construct uninorms via un inorms on bounded lattices, Kybernetika, 60(2)(2024)125–149

    2024

  34. [42]

    Yager, Defending against strategic manipulation in uninorm- based multi-agent decision making, Eur

    R.R. Yager, Defending against strategic manipulation in uninorm- based multi-agent decision making, Eur. J. Operat. Res. 141(1)(2002)217–232

    2002

  35. [43]

    Yager, A

    R.R. Yager, A. Rybalov, Uninorm aggregation operators, Fuzz y Sets Syst. 80(1996)111–120. 22

    1996

  36. [44]

    Zhang, M

    H.P. Zhang, M. Wu, Z. Wang, Y. Ouyang, B. De Baets, A charact erization of the classes Umin and Umax of uninorms on a bounded lattice, Fuzzy Sets Syst. 423(2021)107–121

    2021

  37. [45]

    B. Zhao, T. Wu, Some further results about uninorms on bound ed lattices, Int. J. Approx. Reason. 130(2021)22–49. 23

    2021

Pith tools

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