{
"schema_version": 1,
"kind": "ConceptView",
"producer": "concept-view-1",
"concept_id": "urn:uuid:e5a0d3c6-87bc-459a-b42e-e7d38b8d1601",
"labels": {
"zh-CN": "有序对",
"en": "ordered pair"
},
"alternative_labels": [],
"scope_note": "",
"identity_status": "active",
"identity_revision": "13f746ca9d067df013ac47da578afd4b969c4eab1124565b96bae1805eb4441b",
"occurrences": [
{
"concept_id": "urn:uuid:e5a0d3c6-87bc-459a-b42e-e7d38b8d1601",
"book_id": "szekeres",
"edition_id": "source-snapshot-bc3404156f6e",
"lang": "zh-CN",
"route": "zh/chapter-01/03-products-and-relations",
"anchor": "first-5a0640df55285c563005-zh",
"chapter": "1",
"section": "1.3",
"offset": 370,
"range": [
4560,
4796
],
"source_revision": "45666ab5cdd247b9b5146e4f9df67d1578c145c21402219499435dac061d8141",
"source_category": "translation",
"review_status": "unreviewed",
"role": "occurrence",
"role_basis": {
"kind": "explicit-source-uuid-marker"
},
"label": "有序对(ordered pair)",
"context": {
"range": [
4532,
4830
],
"quote": "集合没有元素的次序：$\\{a,b\\}=\\{b,a\\}$。有序对(ordered pair) $(a,b)$ 则要保留第一、第二个元素的信息。一种纯集合论编码是",
"quote_sha256": "1f968cd9f138f174053d0736bb9babc7801a68cfabdc6df6bd32eb29a994be9b",
"environment": null,
"excerpt_omitted": false
},
"parallel_readings": [
{
"route": "en/chapter-01/03-products-and-relations",
"lang": "en",
"source_category": "original_transcription",
"basis": "registered-book-edition-section-language-context; not-concept-equivalence"
}
],
"id": "occ-2c65fdf8ca25d891652f05f3"
},
{
"concept_id": "urn:uuid:e5a0d3c6-87bc-459a-b42e-e7d38b8d1601",
"book_id": "woit-qtgr",
"edition_id": "2025-10-20",
"lang": "zh-CN",
"route": "books/woit-qtgr/2025-10-20/zh/chapter-01/03-unitary-group-representations",
"anchor": "first-2c46f30ae6e202d6ded9-zh",
"chapter": "1",
"section": "1.3",
"offset": 6703,
"range": [
18247,
18514
],
"source_revision": "4ee587fc4d72d2916ca8f6bce8080ecd42e2ae3d45d92d067027a95982f00008",
"source_category": "translation",
"review_status": "unreviewed",
"role": "occurrence",
"role_basis": {
"kind": "explicit-source-uuid-marker"
},
"label": "有序对(ordered pair)",
"context": {
"range": [
17462,
19026
],
"quote": "\n> [!definition] 定义 · 群在集合上的作用(group action on a set)\n> \n>\n> 定义（群在集合上的作用）。群 $G$ 在集合 $M$ 上的作用由一个映射给出\n>\n> \n>\n> $$\n> (g, x) \\in G \\times M \\to g \\cdot x \\in M\n> $$\n>\n> \n>\n> 将一个有序对(ordered pair) $(g,x)$（其中 $g\\in G$ 是群元素，$x\\in M$ 是集合元素）映为另一个元素 $g\\cdot x\\in M$，使得\n>\n> \n>\n> \n>\n> $$\n> g _ {1} \\cdot (g _ {2} \\cdot x) = (g _ {1} g _ {2}) \\cdot x\\tag{1.2}\n> $$\n>\n> \n>\n> 和\n>\n> \n>\n> $$\n> e \\cdot x = x\n> $$\n>\n> \n>\n> 其中 $e$ 是 $G$ 的恒等元\n",
"quote_sha256": "978f31fc550dd87b0734b2c41f314140bfb5673d236de1fa77e4a4c2e3c8e2db",
"environment": "definition",
"excerpt_omitted": false
},
"parallel_readings": [
{
"route": "books/woit-qtgr/2025-10-20/en/chapter-01/03-unitary-group-representations",
"lang": "en",
"source_category": "original_transcription",
"basis": "registered-book-edition-section-language-context; not-concept-equivalence"
}
],
"id": "occ-1534783016db3b4c0a397b0b"
}
],
"registered_positions": [
{
"route": "zh/chapter-01/03-products-and-relations",
"lang": "zh-CN",
"section": "1.3",
"signal": "term-mark",
"label": "有序对",
"english": "ordered pair",
"offset": 370
},
{
"route": "books/woit-qtgr/2025-10-20/zh/chapter-01/03-unitary-group-representations",
"lang": "zh-CN",
"section": "1.3",
"signal": "term-mark",
"label": "有序对",
"english": "ordered pair",
"offset": 3985
}
],
"statements": [],
"resolved_references": [],
"reading_order": [
"occ-2c65fdf8ca25d891652f05f3",
"occ-1534783016db3b4c0a397b0b"
],
"recorded_dependencies": [],
"reading_paths": [],
"coverage": {
"registered_positions": 2,
"located_occurrences": 2,
"whole_book_search_complete": false,
"registered_positions_truncated": false,
"independent_human_review_implied": false
},
"unresolved": [],
"base_revision": "dddecf22ad7b71685218aae008ac2f0e4162cbcca2c69483141da21a07989981",
"optional_enrichments": [
{
"id": "archived-e5a0d3c6-87bc-459a-b42e-e7d38b8d1601",
"kind": "archived-composition",
"actor": {
"model": "gpt-6-luna",
"reasoning_effort": "high"
},
"source_status": "current",
"stale_reasons": [],
"review_status": {
"generation": "complete",
"structure": "validated",
"semantic_review": "independent-luna-reviewed; human-review-not-performed",
"translation_review": "not-performed; inherited-source-status"
},
"result": {
"overview_claims": [
{
"id": "ordered-pair-order",
"text": "Szekeres 指出无序集合 $\\{a,b\\}=\\{b,a\\}$ 不记录元素次序，并说明有序对 $(a,b)$ 通过指定首项来表达顺序。该节开头的编校说明指出，供给 PDF 印为 $\\{\\{a,b\\},a\\}$，而本节式 (1.1) 是编辑采用的标准 Kuratowski 校正形式：\n\n\n\n$$\n(a,b)=\\{\\{a\\},\\{a,b\\}\\}.\\tag{1.1}\n$$\n\n\n\n作者将这种集合论说明限定为展示“次序”可以如此表达的例子；校正公式不归作 PDF 作者原式。",
"evidence_ids": [
"e-254df14aa95c9081c42b361d"
]
},
{
"id": "ordered-pair-action-use",
"text": "Woit 在群作用的定义中把输入写为 $(g,x)\\in G\\times M$，并将其描述为一个群元素与一个集合元素组成的有序对；这处是使用示例，不是有序对的一般定义。",
"evidence_ids": [
"e-24d6fafc67dd6eef4bdad766"
]
}
],
"author_views": [
{
"id": "szekeres-ordered-pair-view",
"book_id": "szekeres",
"edition_id": "source-snapshot-bc3404156f6e",
"summary": "Szekeres 先指出无序集合 $\\{a,b\\}=\\{b,a\\}$ 不保存顺序，再说明有序对通过指定首项来保留顺序信息。作者称纯集合论表述用于说明“次序”可以如此表达，并说后续无需反复展开该表示。节首编校说明另指出，式 (1.1) 采用标准 Kuratowski 校正，供给 PDF 印式不同；该校正不归于作者原文。",
"conditions": "作者文字讨论二元有序对和指定首项的作用；式 (1.1) 的校正归于节首编校说明，不作为供给 PDF 中作者原式。",
"notation": "$(a,b)$ 表示有序对。节首编校说明指出，本节所列的标准 Kuratowski 公式是校正形式，供给 PDF 印为 $\\{\\{a,b\\},a\\}$。",
"evidence_ids": [
"e-254df14aa95c9081c42b361d"
]
},
{
"id": "woit-ordered-pair-use-view",
"book_id": "woit-qtgr",
"edition_id": "2025-10-20",
"summary": "Woit 在群作用段落把输入写成 $(g,x)\\in G\\times M$，并直接解释它是由群元素 $g$ 和集合元素 $x$ 构成的一对，映到 $g\\cdot x$。这是定义群作用时的记号用法。",
"conditions": "$G$ 是群，$M$ 是其作用所在的集合；该段没有给出有序对的集合论构造。",
"notation": "$(g,x)$ 是作用映射的输入，$G\\times M$ 是其定义域。",
"evidence_ids": [
"e-24d6fafc67dd6eef4bdad766"
]
}
],
"reading_suggestions": [
{
"id": "read-szekeress-1-3",
"text": "阅读 Szekeres 第1.3节中有序对的说明，并同时查看节首编校说明：作者讨论如何通过指定首项记录顺序，式 (1.1) 则是编校采用的 Kuratowski 校正形式，供给 PDF 原印式不同。",
"target_ids": [
"e-254df14aa95c9081c42b361d"
],
"evidence_ids": [
"e-254df14aa95c9081c42b361d"
]
},
{
"id": "read-woit-group-action-use",
"text": "再读 Woit 第1.3节群作用的定义，观察 $(g,x)$ 如何作为映射输入出现；该处展示用法，不取代集合论编码的说明。",
"target_ids": [
"e-24d6fafc67dd6eef4bdad766"
],
"evidence_ids": [
"e-24d6fafc67dd6eef4bdad766"
]
}
],
"assessment": {
"composition": "complete",
"key_material": "available",
"critical_gaps": [],
"whole_book_coverage": "not-exhausted"
},
"gaps": [
"Szekeres 第1.3节的英文转录和中文译文尚未人工核对；Woit 第1.3节的英语转录经过自动比对，仍不足以代替人工核对；所选节选不穷尽全书用法。",
"Szekeres 节首编校说明指出，式 (1.1) 是对供给 PDF 原印式的 Kuratowski 校正；页面将校正与作者原文区分。"
]
},
"evidence": {
"e-a4fec1ce952f578ac31bb927": {
"route": "zh/chapter-01/03-products-and-relations",
"anchor": "",
"statement_id": null,
"label": "Szekeres · §1.3",
"quote_sha256": "ce7f32556d95d76db1005b1c2e6ae7321f1e1a0daef933e69d96d2a7c2ea664f"
},
"e-254df14aa95c9081c42b361d": {
"route": "en/chapter-01/03-products-and-relations",
"anchor": "",
"statement_id": null,
"label": "Szekeres · §1.3",
"quote_sha256": "684291fd921896211fe3bdedefa7b294420cbfda6837bb355dd7308e5b05ff77"
},
"e-be341eb19a86a7ec8fb2e4f8": {
"route": "books/woit-qtgr/2025-10-20/zh/chapter-01/03-unitary-group-representations",
"anchor": "",
"statement_id": null,
"label": "Woit · §1.3",
"quote_sha256": "442a726e83ba789616acffdd77b4002cad1637e3417c11fd0d97710c8fcd9743"
},
"e-24d6fafc67dd6eef4bdad766": {
"route": "books/woit-qtgr/2025-10-20/en/chapter-01/03-unitary-group-representations",
"anchor": "",
"statement_id": null,
"label": "Woit · §1.3",
"quote_sha256": "b0dbd5b12ec734e2daa9ea31c3e9bb4a2153d9f1637924c1d0ba719d6e6a3af6"
}
},
"candidate_dependencies": [],
"input_sha256": "2960ad44dae5aba91d9340ff6cca008d4877f68bcf9a92c5c251f6c05afc862f",
"archive_sha256": "e351808afc593f473729bf2e0cf977ac033493bf4cffe670967a0bac31b87ddb"
}
]
}