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ICFCA 2019: Formal Concept Analysis pp 130-151

# Lattices of Orders

• Christian Meschke
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11511)

## 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.

## References

1. 1.
Dushnik, B., Miller, E.W.: Partially ordered sets. Am. J. Math. 63(3), 600–610 (1941)
2. 2.
Yanagimoto, T., Okamoto, M.: Partial orderings of permutations andmonotonicity of a rank correlation statistic. Ann. Inst. Stat. Math. 21, 489–506 (1969)
3. 3.
Trotter, W.T.: Combinatorics and Partially Ordered Sets. The Johns Hopkins University Press, Baltimore (1992)
4. 4.
Avann, S.P.: The lattice of natural partial orders. Aequationes Math. 8, 95–102 (1972)
5. 5.
Pouzet, M., Reuter, K., Rival, I., Zaguia, N.: A generalized permutahedron. Algebra Univers. 34, 496–509 (1995)
6. 6.
Ganter, B., Meschke, C., Mühle, H.: Merging ordered sets. In: Valtchev, P., Jäschke, R. (eds.) ICFCA 2011. LNCS (LNAI), vol. 6628, pp. 183–203. Springer, Heidelberg (2011).
7. 7.
Ganter, B., Wille, R.: Formal Concept Analysis. Springer, Heidelberg (1999).
8. 8.
Davey, B.A., Priestley, H.A.: Introduction to Lattices and Order, 2nd edn. Cambridge University Press, Cambridge (2002)
9. 9.
Edelman, P.H., Jamison, R.E.: The theory of convex geometries. Geom. Dedicata 19, 247–270 (1985)
10. 10.
Edelman, P.H., Saks, M.E.: Combinatorial representation and convex dimension of convex geometries. Order 5, 23–32 (1988)

## Copyright information

© Springer Nature Switzerland AG 2019

## Authors and Affiliations

1. 1.DresdenGermany