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For the acting group $G=SL_2(\\mathbb{R})^\\mathbb{N}$, we construct a metric minimal PI $G$-flow with $\\mathfrak{c}$ minimal left ideals. We then use this example and results established in \\cite{GW-79} to construct a metric minimal PI cascade $(X,T)$ with $\\mathfrak{c}$ minimal left ideals."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1801.03377","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2018-01-10T13:55:48Z","cross_cats_sorted":[],"title_canon_sha256":"db12e6d727174a60f70e917ee61a4935a0e076b486d86ea361087f49e49179a5","abstract_canon_sha256":"5182a082926742afc5619b336e4c8ce76cd57dc2db1c6bf59908bb8cdbf6e6c4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:22:02.353892Z","signature_b64":"t+6qM6CsSQHVPX5iXlJr/AHPpyVui4+UF1gY232UM4GM1LChIzVV278wRnmFuSiuOzsX5QCy+ER9ItSv8gidAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"922ff237a7933cd9f0461e1dbbe466ddc7f63e71f8894b2b70608c79449c44e2","last_reissued_at":"2026-05-18T00:22:02.353223Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:22:02.353223Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A minimal PI cascade with $2^{\\mathfrak{c}}$ minimal ideals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Eli Glasner, Yair Glasner","submitted_at":"2018-01-10T13:55:48Z","abstract_excerpt":"We first improve an old result of McMahon and show that a metric minimal flow whose enveloping semigroup contains less than $2^{\\mathfrak{c}}$ (where ${\\mathfrak{c}} ={2^{\\aleph_0}}$) minimal left ideals is PI. Then we show the existence of various minimal PI flows with many minimal left ideals, as follows. For the acting group $G=SL_2(\\mathbb{R})^\\mathbb{N}$, we construct a metric minimal PI $G$-flow with $\\mathfrak{c}$ minimal left ideals. We then use this example and results established in \\cite{GW-79} to construct a metric minimal PI cascade $(X,T)$ with $\\mathfrak{c}$ minimal left ideals."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.03377","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1801.03377","created_at":"2026-05-18T00:22:02.353325+00:00"},{"alias_kind":"arxiv_version","alias_value":"1801.03377v3","created_at":"2026-05-18T00:22:02.353325+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1801.03377","created_at":"2026-05-18T00:22:02.353325+00:00"},{"alias_kind":"pith_short_12","alias_value":"SIX7EN5HSM6N","created_at":"2026-05-18T12:32:53.628368+00:00"},{"alias_kind":"pith_short_16","alias_value":"SIX7EN5HSM6NT4CG","created_at":"2026-05-18T12:32:53.628368+00:00"},{"alias_kind":"pith_short_8","alias_value":"SIX7EN5H","created_at":"2026-05-18T12:32:53.628368+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/SIX7EN5HSM6NT4CGDYO3XZDG3X","json":"https://pith.science/pith/SIX7EN5HSM6NT4CGDYO3XZDG3X.json","graph_json":"https://pith.science/api/pith-number/SIX7EN5HSM6NT4CGDYO3XZDG3X/graph.json","events_json":"https://pith.science/api/pith-number/SIX7EN5HSM6NT4CGDYO3XZDG3X/events.json","paper":"https://pith.science/paper/SIX7EN5H"},"agent_actions":{"view_html":"https://pith.science/pith/SIX7EN5HSM6NT4CGDYO3XZDG3X","download_json":"https://pith.science/pith/SIX7EN5HSM6NT4CGDYO3XZDG3X.json","view_paper":"https://pith.science/paper/SIX7EN5H","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1801.03377&json=true","fetch_graph":"https://pith.science/api/pith-number/SIX7EN5HSM6NT4CGDYO3XZDG3X/graph.json","fetch_events":"https://pith.science/api/pith-number/SIX7EN5HSM6NT4CGDYO3XZDG3X/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/SIX7EN5HSM6NT4CGDYO3XZDG3X/action/timestamp_anchor","attest_storage":"https://pith.science/pith/SIX7EN5HSM6NT4CGDYO3XZDG3X/action/storage_attestation","attest_author":"https://pith.science/pith/SIX7EN5HSM6NT4CGDYO3XZDG3X/action/author_attestation","sign_citation":"https://pith.science/pith/SIX7EN5HSM6NT4CGDYO3XZDG3X/action/citation_signature","submit_replication":"https://pith.science/pith/SIX7EN5HSM6NT4CGDYO3XZDG3X/action/replication_record"}},"created_at":"2026-05-18T00:22:02.353325+00:00","updated_at":"2026-05-18T00:22:02.353325+00:00"}