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(2) Let $G$ be a regular $\\mathbb{R}$-factorizable paratopological group; then every subgroup $H$ of $G$ is $\\mathbb{R}$-factorizable if and only if $H$ is $z$-embedded in $G$. This result gives out a positive answer to an question of M.~Sanchis and M.~Tkachenko \\cite[Problem 5.3]{ST}."},"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":"1905.09577","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2019-05-23T10:40:36Z","cross_cats_sorted":[],"title_canon_sha256":"8ee460845ce821707f632e04bd65663014d7267cea67cf733eab0bfbde7426b2","abstract_canon_sha256":"cb30a6a18b181762b00da714fbe33a82764264be963a7cda67718e7136f0a43e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:44:06.735183Z","signature_b64":"JFROkuyxWjmV3jjsGvu3G/VVOrcmlKqGgqbml6ZWmP1n8wMkXIcbIn8788CXmw4O4L7ew2REi6g3hYHQlzslAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"25564857298df583011e1a7d08e4b9639e833c50cd2ac3ade1a8fa25bc2364f7","last_reissued_at":"2026-05-17T23:44:06.734690Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:44:06.734690Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The continuous $d$-open homomorphism images and subgroups of $\\mathbb{R}$-factorizabile paratopological groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GN","authors_text":"Li-Hong Xie, Pengfei Yan","submitted_at":"2019-05-23T10:40:36Z","abstract_excerpt":"In this paper, we prove that: (1) Let $f:G\\rightarrow H$ be a continuous $d$-open surjective homomorphism; if $G$ is an $\\mathbb{R}$-factorizabile paratopological group, then so is $H$. Peng and Zhang's result \\cite[Theorem 1.7]{PZ} is improved. (2) Let $G$ be a regular $\\mathbb{R}$-factorizable paratopological group; then every subgroup $H$ of $G$ is $\\mathbb{R}$-factorizable if and only if $H$ is $z$-embedded in $G$. 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