{"id":"d2204cea-e140-45a2-ae37-56ae2425f420","arxiv_id":"2607.21844","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Effective congruences on mosaics are exactly regular sub-mosaics of M×M satisfying two closure conditions, and the category of mosaics is parabelian.","lead":"This paper characterizes which equivalence relations on mosaics—set-valued generalizations of groups—give well-defined quotient structures, and proves the category of mosaics has a parabelian exactness structure. The characterization is a practical recipe for building quotients of hyperstructures and is applied to endomorphisms and small groups.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.9's necessity proof asserts that an arbitrary morphism f with kernel pair R is the coequalizer of R, which is false (non-surjective f is not a coequalizer in Set); this is a genuine gap in the printed proof of the central characterization, though a direct kernel-pair argument repairs it.","rationale":"Reader's weakest assumption matches the main proof defect I found. The central theorem is the characterization of effective congruences; its proof's necessity direction relies on an unjustified and false claim about being a coequalizer. However, a direct argument using only the definition of kernel pair and the identity/inverse axioms repairs the step, so this is a fixable proof gap rather than a counterexample to the theorem. The imported [NR25] regularity/unitization results are cited from published work and are not internally contradicted by the text. Secondary issues (e.g., the strictness hypothesis in Theorem 4.7 and the mislabeled strict map in Corollary 3.17) are also present, but they do not affect the main characterization as much as the Theorem 3.9 gap. Conditional acceptance is appropriate: the authors should repair the quoted inference and related hypotheses before the proof is treated as fully verified.","tokens_in":25827,"tokens_out":27237,"duration_ms":245536,"concrete_test":"Re-derive the necessity direction of Theorem 3.9 without invoking the false coequalizer claim. Specifically, starting only from f(x)=f(y) defining R, prove: (i) if y∈x⋆e' with f(e')=e_N then f(y)=f(x); (ii) the [e]-on-the-left case; (iii) for mosaics, f(x^{-1})=f(x)^{-1} implies R is inverse-closed. If all three steps hold, the theorem stands and the error is confined to the quoted sentence; if any step fails (e.g., products f(x)⋆e_N not singletons), then the central characterization is actually false.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"In the necessity direction of Theorem 3.9, after taking an effective congruence R = ker f, the proof says: 'by the discussion in 1.1, f is the coequalizer of r1 and r2 in both SMsc and Set.' This is not a valid consequence: a morphism is the coequalizer of its kernel pair only if it is a regular epimorphism, and many morphisms with the same kernel pair are not surjective (or not short). Since Proposition 3.6's necessity direction is then invoked with the quotient of R, the printed proof of the central characterization is incomplete. The gap is repairable: for y∈x⋆e' with f(e')=e_N, applying f gives f(y)∈f(x)⋆e_N, hence f(y)=f(x), so (x,y)∈R; the inverse case follows similarly, and for mosaics f(x^{-1})=f(x)^{-1}. Thus the theorem's conclusion remains plausible, but the proof as written contains a false load-bearing inference.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies quotients in the categories of mosaics and semimosaics. Its main result (Theorem 1.5, proved as Theorem 3.9 and Corollary 3.10) characterizes effective congruences on a (semi)mosaic M as those regular subobjects R⊆M×M whose underlying equivalence relation ≡ satisfies: if y∈x⋆[e]∪[e]⋆x then y≡x, and for mosaics additionally x≡y implies x^{-1}≡y^{-1}; the quotient is the set-theoretic quotient with hyperoperation q(q^{-1}(a)⋆q^{-1}(b)). The characterization is applied to quotients by endomorphisms and automorphism groups, to a decomposition of equivalences into a kernel submosaic plus an equivalence relation on the nonzero cosets, and to explicit quotients of Z/3Z, Z/5Z, and S3, several of which are non-associative. The paper also proves that Msc and cMsc are parabelian (hence proto-exact with normal monos and epis), while failing Barr exactness, protomodularity, Malcev, and proto-abelian properties.","tokens_in":26097,"tokens_out":21267,"duration_ms":201874,"significance":"The main theorem, if correct, gives a complete and elementary description of all quotient objects of mosaics, a class that includes hypergroups and matroids. This goes beyond the previous hypergroup quotient theory and is directly checkable. The paper also provides useful construction tools (endomorphism quotients, coset decomposition) and concrete computations with operation tables. It is a strength that the main characterization is stated as a simple condition on equivalence relations, not on arbitrary congruences. The proofs make heavy use of the published [NR25] infrastructure (regularity, short/coshort equivalence, unitization pushouts); this is legitimate but means several central claims depend on a substantial external apparatus. The parabelian result is a new structural property for mosaics.","major_comments":[{"comment":"The sentence 'by the discussion in 1.1, f is the coequalizer of r1 and r2 in both SMsc and Set' is not a valid consequence. A morphism is the coequalizer of its kernel pair only if it is a regular epimorphism, and an arbitrary morphism f with kernel pair R need not be surjective (hence is not a coequalizer in Set). Since Proposition 3.6 is then invoked with the quotient of R, the printed proof of the central characterization is incomplete. The gap is repairable: let q:M→M/R be the coequalizer of r1 and r2; because R is effective, q is a regular epimorphism and its kernel pair is R, so q is surjective and is the coequalizer in Set, and Proposition 3.6 applies. Equivalently, one can apply f directly to y∈x⋆e' with f(e')=e_N to get f(y)∈f(x)⋆{e_N}={f(x)}, hence f(y)=f(x). Please revise this step.","section":"Theorem 3.9, necessity direction"}],"minor_comments":[{"comment":"'Remnark' should be 'Remark'.","section":"After Theorem 1.5"},{"comment":"The terms 'strict epimorphism' and 'strict monomorphism' are used, but 'strict' is defined in Section 4.2 as equality of images of products. For f:F2→F1, whether f is strict depends on the free semimosaic construction in [NR25]; if 'strict' is intended to mean 'normal', this conflicts with the terminology. Please clarify.","section":"Corollary 3.17"},{"comment":"The statement says 'absorptive subsemimosaic' but the proof requires a strict absorptive subobject for the cokernel M/L to exist; please adjust the statement to 'strict absorptive'.","section":"Theorem 4.7"},{"comment":"In case (2), the third quotient set '{[(23),(12)],[(23)],[(123)],[(132)]}' appears to be a typo; it should probably list [(13)] instead of the second [(23)].","section":"Example 4.10"},{"comment":"The wording 'define a hyperaddition on R:=M×M' and then 'Then R is a congruence on M' is confusing; consider clarifying that R is the universal relation on M equipped with a non-regular subsemimosaic structure.","section":"Example 3.11"},{"comment":"Several typos: 'semimisoaic' (Definition 3.4), 'b (3.7)' (Proposition 3.6), 'folloing' (Example 4.9), 'divisble' (Example 4.2), 'the the set-theoretic quotient' (Section 3.2).","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The gap in Theorem 3.9 is localized and repairable; the authors should be asked to make the repair explicit in the final version. The manuscript leans heavily on [NR25]; since that paper is published, I see no circularity problem, but the borrowed results (regularity, short/coshort classification, unitization pushouts) carry much of the technical weight. The examples and tables are valuable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read this if you work on hyperstructures or regular categories. The paper gives a complete characterization of effective congruences on mosaics and semimosaics: they are exactly the regular subobjects that are semimosaic/mosaic equivalences. That is new and useful, and the small-group quotient tables in Section 4 are a nice payoff. The parabelianity of Msc is also a genuine result, built in a mostly careful way on the [NR25] infrastructure. The authors are honest about what they import, and [NR25] is published, so the reliance is not a circularity problem.\n\nThe main soft spot is Theorem 3.9's necessity proof. The printed argument says an arbitrary morphism f with kernel pair R is the coequalizer of R \"by the discussion in 1.1.\" That is only true for regular epimorphisms, and the stress-test note is correct: the step is false as written. It is also easily repaired — apply f to x⋆e' and use the identity axiom to get f(x)=f(y) — so the theorem itself is not in danger. The authors should fix that line, and also correct Corollary 3.17, which labels f a strict epimorphism when it is only a regular (short) epimorphism.\n\nTheorem 4.7's statement needs a strictness hypothesis on L: for semimosaics it should say \"strict absorptive subsemimosaic,\" not just \"absorptive subsemimosaic.\" Without strictness, the cokernel M/L from Lemma 2.9 is not defined. The examples only use subgroups, so they are unaffected, but the theorem as printed is overbroad. The proof's inference that p2 is again a regular epi is actually valid in a regular category, and \"trivial kernel\" is enough for the bijection they want, even though it would not imply p2 is mono.\n\nOverall: the central characterization is credible, checkable, and an advance over what [NR25] left open. The gaps are localized and repairable. This deserves a serious referee — the request should be for repairs, not rejection.","headline":"Solid extension of [NR25]: the effective-congruence characterization and parabelianity result are real advances, but Theorem 3.9's proof contains a false coequalizer inference and Theorem 4.7 is missing a strictness hypothesis.","tokens_in":26591,"tokens_out":9518,"would_cite":true,"duration_ms":91990,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["18A32","18E08","20N20"],"pacs":[],"model":"deepseek-v4-flash","headline":"Two closure rules describe every quotient of a mosaic","keywords":["mosaics","semimosaics","hypergroups","effective congruences","quotient objects","regular categories","proto-exact categories","parabelian categories"],"falsifier":"Find an equivalence relation satisfying conditions (i) and (ii) whose set-theoretic quotient with the hyperoperation [a]⋆[b]=q(q^{-1}(a)⋆q^{-1}(b)) fails to be a mosaic, or find an effective congruence failing (i) or (ii); the non-effective relation on Z in Example 3.11, with its four-case hyperoperation, is a concrete test case.","tokens_in":25712,"feed_emoji":"🧩","tokens_out":8194,"duration_ms":76375,"temperature":0.7,"pith_summary":"The paper aims to give a complete, checkable description of all quotient objects of mosaics and semimosaics—hyperstructures that generalize hypergroups by dropping associativity while keeping an identity and reversible multiplication. Its central result is that the effective congruences, the equivalence relations that arise as kernel pairs of a morphism, are exactly those satisfying two conditions: multiplying an element by any identity-class element stays in its class, and inversion respects the equivalence. For any such relation, the quotient is the set-theoretic quotient equipped with the hyperoperation [a]⋆[b] = q(q⁻¹(a)⋆q⁻¹(b)), so a quotient exists exactly when the relation passes this test. This matters because earlier treatments of hypergroup quotients did not fully control which quotients existed; the characterization makes existence a purely combinatorial check, and it is applied to quotients by endomorphisms, by automorphism groups, and to explicit quotient mosaics of Z/3Z, Z/5Z, and S₃. The paper also establishes that the category of mosaics is parabelian—its normal monos and normal epis form a proto-exact structure—while failing stronger exactness properties such as Barr exactness and protomodularity.","feed_headline":"Two closure rules describe every quotient of a mosaic","feed_subtitle":"A quotient exists exactly when identity-class products stay in the class and inversion is preserved—a checkable test.","key_machinery":"The central mechanism is the identification of effective congruences with regular sub(semi)mosaics R ⊆ M×M whose underlying set is a (semi)mosaic equivalence relation. The named object is the quotient hyperoperation a⋆b = q(q^{-1}(a)⋆q^{-1}(b)), the unique hyperoperation making the quotient map short. The 'short' morphism condition—p(x)⋆p(y) = p(p^{-1}(x)⋆p^{-1}(y))—is the bridge between categorical regularity and the elementary two conditions of Theorem 1.5; 'coshort' is its monomorphism dual. For the parabelian result, the unitization construction (adjoining an identity to a hypermagma while forcing a subset to become trivial) computes pushouts along normal maps and is used to show normal","core_discovery":"Theorem 1.5 states that for a mosaic M, isomorphism classes of effective congruences on M biject with equivalence relations ≡ on M satisfying (i) if x ∈ y'⋆e' ∪ e'⋆y' for some y'≡y and e'≡e, then x≡y; and (ii) if x≡y then x^{-1}≡y^{-1}. The quotient is the set-theoretic quotient of M by ≡, with the hyperoperation a⋆b = q(q^{-1}(a)⋆q^{-1}(b)). For semimosaics, condition (ii) is omitted. The proof identifies effective congruences with regular subobjects of M×M whose underlying relations are (semi)mosaic equivalences, relying on the previously established fact that in these categories regular epimorphisms are precisely the short surjections and regular monomorphisms the coshort injections; from","pith_inferences":["The relation test turns quotient classification of finite mosaics into a finite saturation computation, so the small-group tables can be extended to larger groups; automating this could give systematic evidence on the paper's closing question of which total mosaics are regular images of groups.","Because the two conditions refer only to products with identity-class elements and inversion, the same characterization may transfer to other regular reversible hyperstructure categories, such as hyperrings, whenever the regular-epi-is-short correspondence holds.","The kernel-plus-punctured-quotient decomposition of Theorem 4.7 suggests a recursive description of quotients: first collapse a normal subobject, then identify the remaining classes arbitrarily; iterating this may generate all quotients without constructing coequalizers."],"forward_implications":["Existence of a quotient is decidable from the relation: checking the two conditions of Theorem 1.5 requires only products with identity-class elements and inversion, not the construction of a cokernel.","Every morphism of (semi)mosaics factors uniquely as a quotient by an effective congruence followed by an injective morphism, giving a complete image factorization.","Every endomorphism φ of a mosaic induces a quotient under x≡y iff φ^m(x)=φ^n(y) for some m,n≥0, and any automorphism-group action induces a quotient by its orbits.","For a group G, all quotient mosaics are obtained from pairs (L,≡) where L is a subgroup and ≡ is an inverse-preserving equivalence relation on the non-trivial double-coset quotient G//L; explicit tables for Z/5Z and S₃ include non-associative mosaics.","The categories Msc and cMsc are parabelian, hence proto-exact with normal monos and normal epis as admissible classes; they are not Barr exact, protomodular, Malcev, or proto-abelian."],"fun_headline_variants":["Two closure rules define every mosaic quotient","Quotient test for mosaics: two conditions only","Mosaic quotients boil down to two equivalence rules","Every mosaic quotient satisfies two simple rules","Characterizing mosaic quotients: two key relations"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The central claim depends on previous work establishing that the categories of mosaics and semimosaics are regular categories with the expected subobject and quotient descriptions, and on a step in the necessity part of Theorem 3.9 that treats a morphism as the coequalizer of its kernel pair—a step that is not valid for arbitrary morphisms, though the intended conclusion can be reached by a direct argument.","fun_headline_variants_meta":{"raw":{"variants":["Two closure rules define every mosaic quotient","Quotient test for mosaics: two conditions only","Mosaic quotients boil down to two equivalence rules","Every mosaic quotient satisfies two simple rules","Characterizing mosaic quotients: two key relations"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000177,"raw_usage":{"total_tokens":1086,"prompt_tokens":657,"completion_tokens":429,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":401,"completion_tokens_details":{"reasoning_tokens":359}},"tokens_in":401,"tokens_out":429,"duration_ms":5034,"temperature":1.0,"reasoning_tokens":359,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T06:33:51.742275+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find an equivalence relation satisfying conditions (i) and (ii) whose set-theoretic quotient with the hyperoperation [a]⋆[b]=q(q^{-1}(a)⋆q^{-1}(b)) fails to be a mosaic, or find an effective congruence failing (i) or (ii); the non-effective relation on Z in Example 3.11, with its four-case hyperoperation, is a concrete test case.","supporting_citations":[],"review_version":1}