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We describe Green's relations on the semigroups $\\mathscr{B}(G)$, $\\mathscr{B}^+(G)$, $\\bar{\\mathscr{B}}(G)$ and $\\bar{\\mathscr{B}}\\,^+(G)$, their bands and show that they are simple, and moreover the semigroups $\\mathscr{B}(G)$ and $\\mathscr{B}^+(G)$ are bisimple. We show that for a commutative linearly ordered group $G$ all non-trivial congruences "},"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":"1111.2401","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2011-11-10T06:37:44Z","cross_cats_sorted":[],"title_canon_sha256":"87f2b80ec474206302e14557163e4e0901d51cd17e1d32a6de0988237d950a9b","abstract_canon_sha256":"d74369f84869dc7cef5fd23518bde5f057ee60bf4121dcaa1340e4805e1903dc"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:05:26.804292Z","signature_b64":"sfqYC7gWVuvVtTulg1eVa9IfLHrQbtdbFk0R5YqJQNRwtLE/cYllfRbbmbMV2hpQ50AjLlaG+lfkRncVpMN/Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"37b0e84f6cdc05725c09ce7946d7064e6a76a8e86020fbab87dff0f69c79a4da","last_reissued_at":"2026-05-18T04:05:26.803617Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:05:26.803617Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Congruences on bicyclic extensions of a linearly ordered group","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Du\\v{s}an Pagon, Kateryna Pavlyk, Oleg Gutik","submitted_at":"2011-11-10T06:37:44Z","abstract_excerpt":"In the paper we study inverse semigroups $\\mathscr{B}(G)$, $\\mathscr{B}^+(G)$, $\\bar{\\mathscr{B}}(G)$ and $\\bar{\\mathscr{B}}\\,^+(G)$ which are generated by partial monotone injective translations of a positive cone of a linearly ordered group $G$. 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