{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:16:05Z","timestamp":1760148965816,"version":"build-2065373602"},"reference-count":25,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2023,6,20]],"date-time":"2023-06-20T00:00:00Z","timestamp":1687219200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"start-up funding for talent","award":["16002203"],"award-info":[{"award-number":["16002203"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>For a quantale I, which is a unit interval endowed with a continuous triangular norm and the Barr extension \u03b2\u00afI of the ultrafilter monad to I-Rel, a characterization of the discrete presheaf monad associated to \u03b2\u00afI is given. It is also proved that, when &amp; is the \u0141ucasiewicz triangular norm, the discrete presheaf monad is isomorphic to the saturated prefilter monad, and when &amp; is the product triangular norm, the prime functional ideal monad is isomorphic to a submonad of the discrete presheaf monad.<\/jats:p>","DOI":"10.3390\/axioms12060610","type":"journal-article","created":{"date-parts":[[2023,6,21]],"date-time":"2023-06-21T02:30:51Z","timestamp":1687314651000},"page":"610","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On Discrete Presheaf Monads"],"prefix":"10.3390","volume":"12","author":[{"given":"Gao","family":"Zhang","sequence":"first","affiliation":[{"name":"School of Mathematical Science, Nanjing Normal University, Nanjing 210046, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0614-8984","authenticated-orcid":false,"given":"Shaoqun","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Intelligence Science and Technology, Nanjing University, Nanjing 210023, China"}]}],"member":"1968","published-online":{"date-parts":[[2023,6,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Mac Lane, S. 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