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Different Differences in Semirings Suciu, Dan Semirings K-relations Relational algebra operates over relations under either set semantics or bag semantics. In 2007 Val Tannen extended the semantics of relational algebra to K-relations, where each tuple is annotated with a value from a semiring. However, only the positive fragment of the relational algebra can be interpreted over K-relations. The reason is that a semiring contains only the operations addition and multiplication, and does not have a difference operation. This paper explores three ways of adding a difference operator to a semiring: as a freely generated algebra, by using the natural order, or by an explicit construction using products and quotients. The paper consists of both a survey of results from the literature, and of some novel results. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Dan Suciu 2024 Is Part Of OASIcs, Volume 119, The Provenance of Elegance in Computation - Essays Dedicated to Val Tannen (2024) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/OASIcs.Tannen.10 urn:nbn:de:0030-drops-201062 https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.Tannen.10 eng https://creativecommons.org/licenses/by/4.0/legalcode