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Lattice Theory of Generalized Partitions

Published online by Cambridge University Press:  20 November 2018

Juris Hartmanis
Juris Hartmanis*
Affiliation:
The Ohio State University
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In (1) the lattice of all equivalence relations on a set S was studied and many important properties were established. In (2) and (3) the lattice of all geometries on a set S was studied and it was shown to be a universal lattice which shares many properties with the lattice of equivalence relations on S. In this paper we shall give the definition of a partition of type n and investigate the lattice formed by all partitions of type n on a fixed set S. It will be seen that a partition of type one on S can be considered as an equivalence relation on S and similarly a partition of type two on S can be considered as a geometry on S as defined in (2). Thus we shall obtain a unified theory of lattices of equivalence relations, lattices of geometries and partition lattices of higher types.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 11 , 1959 , pp. 97 - 106
Copyright
Copyright © Canadian Mathematical Society 1959

References

1. Ore, Oystein, Theory of Equivalence Relations, Duke Math J., 9 (1942), 573–627.Google Scholar
2. Hartmanis, Juris, Two Embedding Theorems for Finite Lattices, Proc. Amer. Math. Soc, 7 (1956), 571-7.Google Scholar
3. Hartmanis, Juris, A Note on the Lattice of Geometries, Proc. Amer. Math. Soc, 8 (1957), 560-2.Google Scholar
4. Birkoff, G., Lattice Theory (New York, 1948).Google Scholar