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We characterize the preorder $\\preccurlyeq _c$, the corresponding equivalence relation of bi-condensability, ${\\mathbb X} \\sim _c {\\mathbb Y}$, and the reversibility of $L$-structures in terms of back and forth systems and the corresponding games. In a similar way we characterize the ${\\mathcal P}_{\\infty \\omega}$-equivalence (which is equi"},"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":"1807.00338","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2018-07-01T14:46:15Z","cross_cats_sorted":[],"title_canon_sha256":"19da19031d54e96ae63f4f685bbc68df47edd435dbb687cfc275c0c0771a0ce6","abstract_canon_sha256":"ffab5ec95ff0177deea18531a04d1aa02d96516e085df0db52e26159c8cb3a4e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:11:55.171234Z","signature_b64":"83Mt9rHQc2kvhaKWjXq/5+dFrXO111GqyXuZe9uqgi3VTFxgBf6S02b0Gy7JHeAPor+kA/zkuW0aK0MyrzYCAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5387681ae332b4ab598bf4775f2a749cb4bce9fc4693194ca381b9eef41eb520","last_reissued_at":"2026-05-18T00:11:55.170349Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:11:55.170349Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Back and Forth Systems of Condensations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Milo\\v{s} S. 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