{ "format": "sumire-research-record", "version": 1, "exportedAt": "2026-11-04T08:27:10.554Z", "scope": "Reader-visible project fields and native live notes only; referenced notes in other projects, deleted notes and private AI context are excluded.", "submission": "not_submitted", "mathematicalCorrespondence": "not_automatically_checked", "project": { "id": "2609.00007", "url": "https://sumire.aiformath.jp/p/2609.00007", "title": "Rieg theory", "tagline": "Semirings with exponentiation, connecting arithmetic, logic, and categorical counting.", "visibility": "public", "owner": { "name": "hora", "handle": "ryuya" }, "origin": "", "source": "kobin:u_EMlmNonOMJFd:slogan:riegs", "goal": "Develop a coherent theory of semirings with exponentiation and use it to connect finite and profinite arithmetic, logic, enumerative combinatorics, and category theory.", "narrative": "A rieg is a commutative semiring equipped with exponentiation satisfying the usual exponent laws, including \\(x^0=1\\) and hence \\(0^0=1\\). This project studies the structure itself and the examples it connects.\n\nThe research programme is described in [note 7](https://sumire.aiformath.jp/p/2609.00007/n/7). The current questions concern [the subrieg generated by nonnegative rationals](https://sumire.aiformath.jp/p/2609.00007/n/1), [game interpretations of the free rieg](https://sumire.aiformath.jp/p/2609.00007/n/3), [functions represented by unary rieg terms](https://sumire.aiformath.jp/p/2609.00007/n/4), and [finite-valued presheaf categories](https://sumire.aiformath.jp/p/2609.00007/n/5).\n\nOne concrete result narrows the topological problem: [a compact Hausdorff topological rieg with cancellative addition is trivial](https://sumire.aiformath.jp/p/2609.00007/n/9). The question [whether a nontrivial connected compact Hausdorff rieg exists](https://sumire.aiformath.jp/p/2609.00007/n/2) therefore reduces to the noncancellative case, where it has a positive answer.\n\nFor finite-valued presheaves, essentially finite slices give a sufficient condition for a topos. [A partial answer](https://sumire.aiformath.jp/p/2609.00007/n/5) proposes an exact criterion stated in terms of finite presheaves. [The right Kan extension approach](https://sumire.aiformath.jp/p/2609.00007/n/8) is absorbed into that note, with the coreflection \\(X\\mapsto X_{\\mathrm{fin}}\\) onto locally finite presheaves as its comonad.\n\nThe next step is concrete: give a criterion for finite presheaf topoi in terms of the indexing category alone.", "history": "", "dates": { "inspiration": null, "sourceRecorded": null, "evidence": null, "registeredInSumire": null, "updatedInSumire": "2026-10-30T14:34:57.980Z", "legacyRecordedAt": "2026-03-30T00:00:00.000Z", "legacyUpdatedAt": "2026-10-30T14:34:57.980Z" }, "goals": [], "goalDependencies": [] }, "notes": [ { "id": "n_W9UnCZv-g0q8", "number": 9, "url": "https://sumire.aiformath.jp/p/2609.00007/n/9", "title": "Compact riegs with cancellative addition are trivial", "author": { "name": "hora", "handle": "ryuya" }, "source": "kobin:u_EMlmNonOMJFd:idea:kw5c85", "body": "## Theorem\n\nLet \\(R\\) be a compact Hausdorff topological rieg. If its additive commutative monoid is cancellative, then \\(R\\) has one element. Connectedness is not required.\n\n## Lemma\n\nEvery compact Hausdorff commutative cancellative topological monoid \\((S,+,0)\\) is a group.\n\n**Proof.** First, every nonempty compact Hausdorff commutative topological semigroup has an idempotent. By compactness and Zorn's lemma, choose a minimal nonempty closed subsemigroup \\(K\\). For \\(a\\in K\\), the set \\(a+K\\) is a nonempty closed subsemigroup (using commutativity), hence equals \\(K\\). Thus\n\\[\nE=\\{x\\in K:a+x=a\\}\n\\]\nis a nonempty closed subsemigroup and equals \\(K\\). In particular, \\(a+a=a\\).\n\nFor \\(s\\in S\\), the closure of \\(\\{ns:n\\geq1\\}\\) is a compact commutative subsemigroup and contains an idempotent \\(e\\). Cancellativity applied to \\(e+e=e+0\\) gives \\(e=0\\). Choose a net \\(n_i s\\to0\\). A subnet of \\((n_i-1)s\\) converges to some \\(t\\), and continuity gives \\(s+t=0\\). Every element therefore has an additive inverse. \\(\\square\\)\n\n## Proof of the theorem\n\nThe lemma gives an additive inverse \\(-1\\) of \\(1\\), and a rieg containing an additive inverse of \\(1\\) is degenerate ([Notes on Rieg Theory](https://sumire.aiformath.jp/p/2609.00007?view=main), Claim 2.3.1: \\(1=0^0=0^{1+(-1)}=0^1\\,0^{-1}=0\\)). \\(\\square\\)\n\n## What remains\n\nThis settles the cancellative case of [the connected compact Hausdorff problem](https://sumire.aiformath.jp/p/2609.00007/n/2). The noncancellative case has a positive answer: the unit interval with \\(\\max\\), \\(\\min\\) and \\(x^y=\\max(x,1-y)\\) is a nontrivial connected compact Hausdorff rieg ([[2609.00007/2]]).", "dates": { "inspiration": null, "sourceRecorded": null, "evidence": null, "registeredInSumire": null, "updatedInSumire": "2026-10-30T14:31:08.891Z", "legacyRecordedAt": "2026-08-27T00:00:00.000Z", "legacyUpdatedAt": "2026-10-30T14:31:08.891Z" }, "language": null, "projectLanguage": "ja", "languageSource": "not_recorded", "legacyStage": "proof", "classification": "unclassified", "classificationStatus": "unclassified", "importance": "unspecified", "section": "main", "problemState": "unknown", "resolution": "", "verificationEvidence": { "ai": "", "lean": "", "human": "" }, "aiClaimModel": "", "formalization": "unknown", "humanReview": "unreviewed", "verification": "unverified", "ai": { "use": "assisted", "tools": "ChatGPT, Codex, Claude", "additions": "" }, "originalQuestion": "", "references": [] } ], "noteDependencies": [], "guides": { "arxiv": "https://info.arxiv.org/help/submit/index.html", "formal_conjectures": "https://github.com/google-deepmind/formal-conjectures/blob/main/CONTRIBUTING.md", "mathlib": "https://leanprover-community.github.io/contribute/index.html" }, "search": { "arxiv": "https://arxiv.org/search/", "formal_conjectures": "https://google-deepmind.github.io/formal-conjectures/browse/", "mathlib": "https://leanprover-community.github.io/mathlib4_docs/" } }