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Our intention is to examine a\u00a0stronger version of this axiom according to which if<jats:bold><jats:italic>x<\/jats:italic><\/jats:bold>is in contact with the supremum of some family<jats:bold><jats:italic>S<\/jats:italic><\/jats:bold>of regions, then there is a\u00a0<jats:bold><jats:italic>y<\/jats:italic><\/jats:bold>in<jats:bold><jats:italic>S<\/jats:italic><\/jats:bold>that is in contact with<jats:bold><jats:italic>x<\/jats:italic><\/jats:bold>. We study a\u00a0modal possibility operator which is definable in complete algebras in the presence of the aforementioned axiom, and we prove that the class of complete algebras satisfying the axiom is closely related to the class of modal KTB-algebras. We also demonstrate that in the class of complete extensional contact algebras the axiom is equivalent to the statement:<jats:italic>every region is isolated<\/jats:italic>. Finally, we present an\u00a0interpretation of the modal operator in the class of the so-called<jats:italic>resolution contact algebras<\/jats:italic>.<\/jats:p>","DOI":"10.1007\/s11225-023-10036-7","type":"journal-article","created":{"date-parts":[[2023,4,26]],"date-time":"2023-04-26T17:03:32Z","timestamp":1682528612000},"page":"717-748","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["From Contact Relations to Modal Operators, and Back"],"prefix":"10.1007","volume":"111","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3379-0577","authenticated-orcid":false,"given":"Rafa\u0142","family":"Gruszczy\u0144ski","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Paula","family":"Mench\u00f3n","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2023,4,26]]},"reference":[{"key":"10036_CR1","doi-asserted-by":"crossref","unstructured":"Bezhanishvili, G., N. 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