Abstract
Ordering order relations is a well-established topic of order theory. Traditionally, the concept of order extension plays an important role and leads to fundamental results like Dushnik and Miller’s Theorem stating that every order is the intersection of linear extensions [1]. We introduce an alternative but still quite elementary way to order relations. The resulting lattices of orders can be viewed as a generalisation of the lattices of permutations from [2] and accordingly maintain a very high degree of symmetry. Furthermore, the resulting lattices of orders form complete sublattices of it’s quasiorder counterpart, the lattices of quasiorders, which are also introduced. We examine the basic properties of these two classes of lattices and present their contextual representations.
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- 1.
This means that for relations A, \(B_t\) (\(t\in T\)) and C the following two distributivity laws hold: \(A\circ (\bigcup _t B_t) = \bigcup _t(A\circ B_t)\) and \((\bigcup _t B_t) \circ C = \bigcup _t(B_t\circ C)\).
- 2.
Please remember that \((\cdot )^\circ \) denotes the transitive closure and that \({{\,\mathrm{\blacktriangleright }\,}}\) denotes the q-saturation introduced in Definition 1.
- 3.
We write \(P\mathrel {\sqsubset _L}Q\) if \(P\mathrel {\sqsubseteq _L}Q\) and \(P\ne Q\).
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Meschke, C. (2019). Lattices of Orders. In: Cristea, D., Le Ber, F., Sertkaya, B. (eds) Formal Concept Analysis. ICFCA 2019. Lecture Notes in Computer Science(), vol 11511. Springer, Cham. https://doi.org/10.1007/978-3-030-21462-3_10
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DOI: https://doi.org/10.1007/978-3-030-21462-3_10
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