{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,17]],"date-time":"2026-02-17T03:36:57Z","timestamp":1771299417866,"version":"3.50.1"},"reference-count":19,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2025,2,1]],"date-time":"2025-02-01T00:00:00Z","timestamp":1738368000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Many information-theoretic quantities have corresponding representations in terms of sets. Many of these information quantities do not have a fixed sign\u2014for example, the co-information can be both positive and negative. In previous work, we presented a signed measure space for entropy where the smallest sets (called atoms) all have fixed signs. In the present work, we demonstrate that these atoms have natural algebraic behaviour which can be expressed in terms of ideals (characterised here as upper sets), and we show that this behaviour allows us to make bounding arguments and describe many fixed-sign information quantity expressions. As an application, we give an algebraic proof that the only completely synergistic system of three finite variables X, Y and Z=f(X,Y) is the XOR gate.<\/jats:p>","DOI":"10.3390\/e27020151","type":"journal-article","created":{"date-parts":[[2025,2,3]],"date-time":"2025-02-03T05:36:32Z","timestamp":1738560992000},"page":"151","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Algebraic Representations of Entropy and Fixed-Sign Information Quantities"],"prefix":"10.3390","volume":"27","author":[{"ORCID":"https:\/\/orcid.org\/0009-0006-3730-0295","authenticated-orcid":false,"given":"Keenan J. A.","family":"Down","sequence":"first","affiliation":[{"name":"Department of Psychology, School of Biological and Behavioural Sciences, Queen Mary University of London, Mile End Road, Bethnal Green, London E1 4NS, UK"},{"name":"Department of Psychology, University of Cambridge, Downing Site, Downing Place, Cambridge CB2 3EB, UK"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1789-5894","authenticated-orcid":false,"given":"Pedro A. M.","family":"Mediano","sequence":"additional","affiliation":[{"name":"Department of Computing, Imperial College London, 180 Queen\u2019s Gate, South Kensington, London SW7 2RH, UK"},{"name":"Division of Psychology and Language Sciences, University College London, 26 Bedford Way, London WC1H 0AP, UK"}]}],"member":"1968","published-online":{"date-parts":[[2025,2,1]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"466","DOI":"10.1109\/18.79902","article-title":"A new outlook on Shannon\u2019s information measures","volume":"37","author":"Yeung","year":"1991","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"439","DOI":"10.1137\/1107041","article-title":"On the amount of information","volume":"7","author":"Ting","year":"1962","journal-title":"Theory Probab. Its Appl."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"James, R.G., and Crutchfield, J.P. (2017). Multivariate dependence beyond Shannon information. Entropy, 19.","DOI":"10.3390\/e19100531"},{"key":"ref_4","unstructured":"Williams, P.L., and Beer, R.D. (2010). Nonnegative decomposition of multivariate information. arXiv."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Kolchinsky, A. (2022). A novel approach to the partial information decomposition. Entropy, 24.","DOI":"10.3390\/e24030403"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Ince, R.A. (2017). Measuring multivariate redundant information with pointwise common change in surprisal. 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A logarithmic decomposition for information. Proceedings of the 2023 IEEE International Symposium on Information Theory (ISIT), Taipei, Taiwan.","DOI":"10.1109\/ISIT54713.2023.10206673"},{"key":"ref_11","unstructured":"Down, K.J., and Mediano, P.A. (2024). A logarithmic decomposition and a signed measure space for entropy. arXiv."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1945","DOI":"10.3390\/e13111945","article-title":"A characterization of entropy in terms of information loss","volume":"13","author":"Baez","year":"2011","journal-title":"Entropy"},{"key":"ref_13","unstructured":"Bell, A.J. (2004, January 22\u201324). The co-information lattice. 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Towards an extended taxonomy of information dynamics via integrated information decomposition. arXiv."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/2\/151\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T16:25:37Z","timestamp":1760027137000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/2\/151"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,2,1]]},"references-count":19,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2025,2]]}},"alternative-id":["e27020151"],"URL":"https:\/\/doi.org\/10.3390\/e27020151","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,2,1]]}}}