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Among these various algebraic structures, for which we employ the umbrella term <jats:italic>intuitionistic modal algebras<\/jats:italic>, some have been studied since at least the 1970s, usually within the framework of topology and sheaf theory. Others may seem more exotic, for their primitive operations arise from algebraic terms of the intuitionistic modal language which have not been previously considered. We shall for instance investigate the variety of <jats:italic>weak implicative semilattices<\/jats:italic>, whose members are (non-necessarily distributive) meet semilattices endowed with a nucleus and an implication operation which is not a relative pseudo-complement but satisfies the postulates of Celani and Jansana\u2019s strict implication. For each of these new classes of algebras we establish a representation and a topological duality which generalize the known ones for Heyting algebras enriched with a nucleus.<\/jats:p>","DOI":"10.1007\/s11225-023-10065-2","type":"journal-article","created":{"date-parts":[[2023,9,15]],"date-time":"2023-09-15T08:03:47Z","timestamp":1694765027000},"page":"611-660","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Intuitionistic Modal Algebras"],"prefix":"10.1007","volume":"112","author":[{"given":"Sergio A.","family":"Celani","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1364-5003","authenticated-orcid":false,"given":"Umberto","family":"Rivieccio","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2023,9,15]]},"reference":[{"key":"10065_CR1","first-page":"357","volume":"12","author":"R Beazer","year":"1986","unstructured":"Beazer, R., Varieties of modal lattices, Houston J. 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