{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,9,6]],"date-time":"2024-09-06T03:49:00Z","timestamp":1725594540779},"publisher-location":"Vienna","reference-count":12,"publisher":"Springer Vienna","isbn-type":[{"type":"print","value":"9783211829714"},{"type":"electronic","value":"9783709165102"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[1997]]},"DOI":"10.1007\/978-3-7091-6510-2_1","type":"book-chapter","created":{"date-parts":[[2011,7,4]],"date-time":"2011-07-04T07:31:40Z","timestamp":1309764700000},"page":"1-21","source":"Crossref","is-referenced-by-count":1,"title":["Background Material"],"prefix":"10.1007","author":[{"given":"Peter","family":"Jipsen","sequence":"first","affiliation":[]},{"given":"Chris","family":"Brink","sequence":"additional","affiliation":[]},{"given":"Gunther","family":"Schmidt","sequence":"additional","affiliation":[]}],"member":"297","reference":[{"key":"1_CR1","unstructured":"Chris Brink gratefully acknowledges the longstanding financial support of the South African Foundation for Research Development."},{"key":"1_CR2","unstructured":"The symbol \u225c signifies that the equalities hold by definition."},{"key":"1_CR3","unstructured":"Since the axioms for lattices, distributivity and complementation are equational."},{"key":"1_CR4","unstructured":"The constant I does so vacuously, since it has no arguments."},{"key":"1_CR5","unstructured":"In categorical terms f is the left adjoint (of g), and g is the right adjoint (of f). To avoid confusion with left and right residuals of relations, we refrain from this terminology."},{"key":"1_CR6","unstructured":"Note that in general (F, \u2291) is not a sublattice of (X, \u2291)."},{"key":"1_CR7","unstructured":"Note that if I = \u03b8 then \u220fi\u2208\u03b8 Bi one-element algebra, hence one-element algebras are not subdirectly irreducible."},{"key":"1_CR8","unstructured":"In fact it is an algebraic lattice since every congruence is the join of the compact (= finitely generated) congruences that it contains."},{"key":"1_CR9","unstructured":"The distinction between set and class is not relevant here, so readers unfamiliar with classes may just think of them as sets."},{"key":"1_CR10","unstructured":"Recall that we have used the small circle rather than the semicolon to indicate functional composition; we do the same for morphisms. It is useful to keep in mind that by convention this inverts the order: g o f = f;g."},{"key":"1_CR11","unstructured":"\u22a8 is called the satisfaction symbol and denotes semantical truth."},{"key":"1_CR12","unstructured":"This traditional terminology is motivated by philosophy; in particular by the Leibnizian idea that \u201cnecessarily true\u201d means \u201ctrue in all possible worlds\u201d."}],"container-title":["Relational Methods in Computer Science"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/978-3-7091-6510-2_1.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,11,23]],"date-time":"2020-11-23T04:45:26Z","timestamp":1606106726000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/978-3-7091-6510-2_1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1997]]},"ISBN":["9783211829714","9783709165102"],"references-count":12,"URL":"https:\/\/doi.org\/10.1007\/978-3-7091-6510-2_1","relation":{},"subject":[],"published":{"date-parts":[[1997]]}}}