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In particular, we prove that for each $\\mathcal{D}$-class $D$ of a graph inverse semigroup over an acyclic graph the semigroup $D^0$ is isomorphic to a semigroup of matrix units. Also we show that for any elements $a,b$ of a graph inverse semigroup $G(E)$, $J_a\\cdot J_b\\cup J_b\\cdot J_a\\subset J_b^0$ if there exists a path $w$ such that $s(w)\\in J_a$ and $r(w)\\in J_b$."},"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":"1806.09671","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2018-06-25T19:18:01Z","cross_cats_sorted":[],"title_canon_sha256":"e1d706e85a73179a055beb5089bd36fada126722d8ed18349eff68364a5e80fd","abstract_canon_sha256":"ac52163fb9ab3328d9ac3d4dec261afb14d5fb87a8b03ee12879d60253651d11"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:48:06.634380Z","signature_b64":"S8Y8xiGGNNimZx/9lhmy7IG5g5ZusFjHw4gSuM2U17X1OmWzMcXG2LGlos5u6xqEIPQnqE5hJNnGEStBJmI3Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5ca30cdd34ca0c88483091a120b0a70df5a81730e9b7c4bfd5fe8cc999f533fb","last_reissued_at":"2026-05-17T23:48:06.633999Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:48:06.633999Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"An alternative look at the structure of graph inverse semigroups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Serhii Bardyla","submitted_at":"2018-06-25T19:18:01Z","abstract_excerpt":"For any graph inverse semigroup $G(E)$ we describe subsemigroups $D^0=D\\cup\\{0\\}$ and $J^0=J\\cup\\{0\\}$ of $G(E)$ where $D$ and $J$ are arbitrary $\\mathcal{D}$-class and $\\mathcal{J}$-class of $G(E)$, respectively. In particular, we prove that for each $\\mathcal{D}$-class $D$ of a graph inverse semigroup over an acyclic graph the semigroup $D^0$ is isomorphic to a semigroup of matrix units. Also we show that for any elements $a,b$ of a graph inverse semigroup $G(E)$, $J_a\\cdot J_b\\cup J_b\\cdot J_a\\subset J_b^0$ if there exists a path $w$ such that $s(w)\\in J_a$ and $r(w)\\in J_b$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.09671","kind":"arxiv","version":2},"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":"1806.09671","created_at":"2026-05-17T23:48:06.634060+00:00"},{"alias_kind":"arxiv_version","alias_value":"1806.09671v2","created_at":"2026-05-17T23:48:06.634060+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1806.09671","created_at":"2026-05-17T23:48:06.634060+00:00"},{"alias_kind":"pith_short_12","alias_value":"LSRQZXJUZIGI","created_at":"2026-05-18T12:32:37.024351+00:00"},{"alias_kind":"pith_short_16","alias_value":"LSRQZXJUZIGIQSBQ","created_at":"2026-05-18T12:32:37.024351+00:00"},{"alias_kind":"pith_short_8","alias_value":"LSRQZXJU","created_at":"2026-05-18T12:32:37.024351+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/LSRQZXJUZIGIQSBQSGQSBMFHBX","json":"https://pith.science/pith/LSRQZXJUZIGIQSBQSGQSBMFHBX.json","graph_json":"https://pith.science/api/pith-number/LSRQZXJUZIGIQSBQSGQSBMFHBX/graph.json","events_json":"https://pith.science/api/pith-number/LSRQZXJUZIGIQSBQSGQSBMFHBX/events.json","paper":"https://pith.science/paper/LSRQZXJU"},"agent_actions":{"view_html":"https://pith.science/pith/LSRQZXJUZIGIQSBQSGQSBMFHBX","download_json":"https://pith.science/pith/LSRQZXJUZIGIQSBQSGQSBMFHBX.json","view_paper":"https://pith.science/paper/LSRQZXJU","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1806.09671&json=true","fetch_graph":"https://pith.science/api/pith-number/LSRQZXJUZIGIQSBQSGQSBMFHBX/graph.json","fetch_events":"https://pith.science/api/pith-number/LSRQZXJUZIGIQSBQSGQSBMFHBX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/LSRQZXJUZIGIQSBQSGQSBMFHBX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/LSRQZXJUZIGIQSBQSGQSBMFHBX/action/storage_attestation","attest_author":"https://pith.science/pith/LSRQZXJUZIGIQSBQSGQSBMFHBX/action/author_attestation","sign_citation":"https://pith.science/pith/LSRQZXJUZIGIQSBQSGQSBMFHBX/action/citation_signature","submit_replication":"https://pith.science/pith/LSRQZXJUZIGIQSBQSGQSBMFHBX/action/replication_record"}},"created_at":"2026-05-17T23:48:06.634060+00:00","updated_at":"2026-05-17T23:48:06.634060+00:00"}