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It has been used to model various application domains, from signal-flow graphs to Petri nets and electrical circuits. In this paper, we introduce to the family its most expressive member to date: Graphical Piecewise-Linear Algebra, a new language to specify piecewise-linear subsets of vector spaces.<\/jats:p><jats:p>Like the previous members of the family, it comes with a complete axiomatisation, which means it can be used to reason about the corresponding semantic domain purely equationally, forgetting the set-theoretic interpretation. We show completeness using a single axiom on top of Graphical Polyhedral Algebra, and show that this extension is the smallest that can capture a variety of relevant constructs.<\/jats:p><jats:p>Finally, we showcase its use by modelling the behaviour of stateless electronic circuits of ideal elements, a domain that had remained outside the remit of previous diagrammatic languages.<\/jats:p>","DOI":"10.1007\/978-3-030-99253-8_6","type":"book-chapter","created":{"date-parts":[[2022,3,28]],"date-time":"2022-03-28T20:02:48Z","timestamp":1648497768000},"page":"101-119","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Graphical Piecewise-Linear Algebra"],"prefix":"10.1007","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5244-893X","authenticated-orcid":false,"given":"Guillaume","family":"Boisseau","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3945-2704","authenticated-orcid":false,"given":"Robin","family":"Piedeleu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2022,3,29]]},"reference":[{"key":"6_CR1","unstructured":"Baez, J.C., Coya, B., Rebro, F.: Props in network theory. 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