{"id":"5185c29e-0717-4bfa-b977-b5a37daa890b","arxiv_id":"2505.04220","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Two comparable closure operators on a bounded lattice yield new uninorms whose values in the cross-region depend on the chosen operators, under necessary and sufficient conditions.","lead":"This paper gives new recipes for building uninorms, associative operations used in fuzzy logic, on partially ordered structures called bounded lattices. The recipes use two coordinated closure operators and produce operations outside the previously known standard families.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No flaw found in Theorem 3.1's proof, but Example 3.1's Table 1 is invalid: cl2 violates closure axiom CL2.","rationale":"The reader's weakest-assumption pick, comparability of cl1 and cl2, is explicitly stated in Theorem 3.1 and is used in a correct way in the monotonicity proof; it does not look like a hidden gap. I therefore looked for another load-bearing weakness. The main theorem's proof appears internally consistent: the necessity arguments use only associativity with 0 and e, the sufficiency proof's monotonicity cases are checkable, and the associativity cases match Proposition 2.3 exactly. The concrete defect I found is in the supporting Example 3.1: Table 1's cl2 fails CL2 because a∨e=e but cl2(a)∨cl2(e)=j∨e=j≠e. This invalidates the example as evidence for the construction, but it does not invalidate the theorem. Since the central claim itself stands and the reader's CONDITIONAL verdict already reflects the need for presentation fixes, I leave the verdict unchanged.","tokens_in":21080,"tokens_out":28325,"duration_ms":275118,"concrete_test":"Recompute Table 1 against CL2: for each a ∈ ]0,e[, CL2 forces cl2(a)∨cl2(e)=cl2(e), and if cl2(e)=e then cl2(a)≤e. Any corrected table must satisfy these equations for all joins; after replacing cl2 with a genuine closure operator satisfying cl1≤cl2 and the two stated range conditions, regenerate Table 2 and verify that U is still a uninorm.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central proof of Theorem 3.1 survives scrutiny. The necessity arguments are sound, the monotonicity subcase 1.2.3 does use the comparability assumption but in a valid way (cl1(x) ≤ cl2(x) ≤ cl2(y) and e ≤ y∨e imply the required meet inequality), and the associativity partition via Proposition 2.3 covers all combinations of the sets ]0,e[, Ie, and ]e,1]. The real concrete defect is in Example 3.1. In Table 1, a ∈ ]0,e[ and j ∈ ]e,1], with cl2(a)=j and cl2(e)=e. Closure axiom CL2 requires cl2(a∨e)=cl2(a)∨cl2(e). Since a<e, a∨e=e, so the left side is cl2(e)=e, while the right side is j∨e=j, because j>e. Thus e=j, a contradiction. The same failure occurs for b. Hence cl2 is not a closure operator, and Table 2 is not an instance of the theorem as stated. This is an illustrative and evidential flaw rather than a refutation of the main theorem: identity closures satisfy the hypotheses trivially, and the proof itself does not depend on this example. Still, the paper's only displayed non-degenerate pair of closure operators is invalid and should be corrected.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":21338,"tokens_out":19856,"duration_ms":168854,"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":[{"comment":"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.","section":"Example 3.1 (Tables 1 and 2)"},{"comment":"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.","section":"Theorems 3.2 and 3.4"},{"comment":"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.","section":"Theorem 3.3 proof"}],"minor_comments":[{"comment":"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.","section":"Theorem 3.1, monotonicity case 1.2.3"},{"comment":"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.","section":"Proposition 3.1"},{"comment":"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.","section":"Example 3.3 (Tables 4 and 5)"}],"recommendation":"major_revision","confidential_remarks":"The central Theorem 3.1 appears sound, and the overall strategy is worth publishing if the presentation is repaired. However, the tabular examples currently do not satisfy the definitions, and the dual theorems are unproved; these issues are visible to any careful reader and should be fixed before the paper can be accepted. I would also encourage the authors to have the tables machine-checked, since the displayed operations in Examples 3.1 and 3.3 do not match the formulas that precede them."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"One thing you should know: the main construction is real and the proof of Theorem 3.1 holds together, but the paper's only non-trivial worked example is invalid, and the dual theorems are asserted without proof. Neither defect kills the central result, but both need attention before this is publication-ready.\n\nWhat's new: prior closure-operator constructions of uninorms on bounded lattices all land in U*_min ∪ U^1_min or the dual classes. Xiu and Zheng use two comparable closure operators (or two comparable interior operators) and get values on the cross region that depend on the operators, so the resulting uninorms can fall outside those classes. The necessary and sufficient conditions on cl1 and cl2 look right. I checked the monotonicity and associativity cases in Theorem 3.1; the comparability assumption is used genuinely in case 1.2.3, and the associativity partition through Proposition 2.3 covers all combinations. The degenerate single-operator versions and the recovery of the Karaçal–Mesiar construction are a nice consistency check.\n\nThe soft spots are real. The stress test is correct: Example 3.1's cl2 is not a closure operator. Since a < e, a∨e = e, so CL2 requires cl2(e) = cl2(a) ∨ cl2(e), i.e. e = j∨e = j, impossible. So Table 1 and Table 2 do not instantiate the theorem. The conclusion explicitly points to Tables 1 and 4 as evidence that non-trivial operators exist; Table 4 should be checked with the same care, though the stress test did not flag it. Also, Theorems 3.2 and 3.4 are stated without proof. For a construction paper that is a real gap, even if the duality is plausible. Several monotonicity subcases are dismissed as 'easy to see'; that is acceptable in a short paper but should be expanded or consolidated into a lemma. One minor technical point: in the necessity proof for cl2 in Theorem 3.1, the claim that cl2(x) ∧ (x∨e) ∈ ]e,1] is not automatic — the meet can be e — but the contradiction only needs the value to be nonzero, so the argument survives with a line changed.\n\nBottom line: this is a competent, narrow contribution to the uninorm-on-lattices literature. The main theorem appears correct and new. The example bug is embarrassing but fixable. I'd send it to a knowledgeable referee and ask for corrected examples plus at least sketches of the dual proofs. The authors should also double-check Example 3.3.","headline":"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.","tokens_in":21860,"tokens_out":5479,"would_cite":true,"duration_ms":48362,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["06B05","03E72"],"pacs":[],"model":"deepseek-v4-flash","headline":"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].","keywords":["bounded lattices","uninorms","closure operators","interior operators","t-conorms","t-norms","neutral element","piecewise constructions"],"falsifier":"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.","tokens_in":20872,"feed_emoji":"🧩","tokens_out":12336,"duration_ms":119702,"temperature":0.7,"pith_summary":"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.","feed_headline":"Two closure operators build uninorms outside usual classes","feed_subtitle":"The construction makes mixed-region outputs depend on the chosen operators, not just the lattice.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Introduces uninorms on bounded lattices and supplies the baseline $U_s$ and $U_t$ constructions that the identity case of the new recipe reproduces.","marker":"[30]"},{"why":"Defines closure operators on lattices, the maps whose inflation, idempotence, and monotonicity the construction relies on.","marker":"[20]"},{"why":"Provides the definition of a t-conorm, the operation used on the upper interval $[e,1]$ in Theorems 3.1 and 3.3.","marker":"[15]"},{"why":"Defines the classes $U_{\\min}^{*}$, $U_{\\min}^{1}$, $U_{\\max}^{*}$, and $U_{\\max}^{0}$ that frame the claim that the new uninorms need not belong to the standard families.","marker":"[44]"},{"why":"Supplies the block-wise associativity criterion used in Proposition 2.3 to reduce the associativity proof to finitely many region cases.","marker":"[28]"},{"why":"Presents prior single-closure-operator uninorm constructions whose mixed-region values were fixed by the lattice, serving as the contrast case.","marker":"[34]"},{"why":"Catalogs earlier sufficient-and-necessary closure-operator conditions on bounded lattices that the two-operator results extend.","marker":"[9]"}],"fun_headline_variants":["Two closures yield uninorms beyond standard classes","New uninorm construction via comparable closure operators","Uninorms from dual operators escape known families","Closure operators pair up for fresh uninorm structures","Bounded lattices: two operators, new uninorms"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Two closures yield uninorms beyond standard classes","New uninorm construction via comparable closure operators","Uninorms from dual operators escape known families","Closure operators pair up for fresh uninorm structures","Bounded lattices: two operators, new uninorms"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000809,"raw_usage":{"total_tokens":3592,"prompt_tokens":1029,"completion_tokens":2563,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":645,"completion_tokens_details":{"reasoning_tokens":2491}},"tokens_in":645,"tokens_out":2563,"duration_ms":17056,"temperature":1.0,"reasoning_tokens":2491,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T23:34:24.990692+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Kara¸ cal, R","cited_arxiv_id":null,"evidence_quote":"Introduces uninorms on bounded lattices and supplies the baseline $U_s$ and $U_t$ constructions that the identity case of the new recipe reproduces."},{"cited_title":"Everett, Closure operators and Galois theory in lattices, T rans","cited_arxiv_id":null,"evidence_quote":"Defines closure operators on lattices, the maps whose inflation, idempotence, and monotonicity the construction relies on."},{"cited_title":"C ¸ aylı, F","cited_arxiv_id":null,"evidence_quote":"Provides the definition of a t-conorm, the operation used on the upper interval $[e,1]$ in Theorems 3.1 and 3.3."},{"cited_title":"Zhang, M","cited_arxiv_id":null,"evidence_quote":"Defines the classes $U_{\\min}^{*}$, $U_{\\min}^{1}$, $U_{\\max}^{*}$, and $U_{\\max}^{0}$ that frame the claim that the new uninorms need not belong to the standard families."},{"cited_title":"Ji, Constructions of uninorms on bounded lattices by means o f t-subnorms and t-subconorms, Fuzzy Sets Syst","cited_arxiv_id":null,"evidence_quote":"Supplies the block-wise associativity criterion used in Proposition 2.3 to reduce the associativity proof to finitely many region cases."},{"cited_title":"Ouyang, H.P","cited_arxiv_id":null,"evidence_quote":"Presents prior single-closure-operator uninorm constructions whose mixed-region values were fixed by the lattice, serving as the contrast case."},{"cited_title":"C ¸ aylı, New construction approaches of uninorms on bounde d lattices, Int","cited_arxiv_id":null,"evidence_quote":"Catalogs earlier sufficient-and-necessary closure-operator conditions on bounded lattices that the two-operator results extend."}],"review_version":1}