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We present a new operator ${ \\mathbf{V} \\mapsto \\mathbf{V}^\\mathsf{bar} }$ that preserves the property of join irreducibility, as does the dual operator, and show "},"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":"1702.03753","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2017-02-13T12:59:29Z","cross_cats_sorted":["cs.FL","math.RA"],"title_canon_sha256":"885098ff0edf04c2743e5ad77e9c22328219b8735d986ef013c8d8d704c24aa5","abstract_canon_sha256":"89fbceea45dc55c2a09019c10c6235366071b7bb64c06c51ba843a8993dbf6a0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:41:56.767080Z","signature_b64":"fhIKu7pDhUvRW+N6PaR152mBvK3YqZNK0MpcmwDiE2WPJ7PfWvM9kkJNeMWdZxZNJda/P7++MhipbHMAa3f0DQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"46126dbc0525d6a6722d6bfcbd34f1db474d73bee65d511735004fbb57f7100e","last_reissued_at":"2026-05-17T23:41:56.766433Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:41:56.766433Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Join irreducible semigroups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.FL","math.RA"],"primary_cat":"math.GR","authors_text":"Benjamin Steinberg, Edmond W. 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