Abstract
In (7.16), a matroid lattice is defined as a compactly atomistic M-symmetric lattice, and it may be defined as an upper continuous AC-lattice. It was shown in (7.10) and (7.15) that in a compactly atomistic lattice (α) the property of being M-symmetric, (β) the covering property, and (γ) the exchange property are equivalent. In DubeilJacotin, Leisieur and Croisot [1] a compactly atomistic lattice with (β) is called a geometric lattice, and in MacLane [1] a compactly atomistic lattice with (γ) is called an exchange lattice.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References for Chapter III
M. L. Dubreil-Jacotin, L. Leisieur and R. Croisot: Leçons sur la théorie des treillis des structures algébriques ordonnées et des treillis géométriques. Paris: Gauthier-Villars 1953.
S. MacLane: A lattice formulation for transcendence degrees and p-bases. Duke Math. J. 4, 455–468 (1938).
F. Maeda: Perspectivity of points in matroid lattices. J. Sci. Hiroshima Univ., Ser. A-128, 101–112 (1964).
U. Sasaki and S. Fujiwara: The decomposition of matroid lattices. J. Sci. Hiroshima Univ., Ser. A 15, 183–188 (1952).
U. Sasaki: Semi-modularity in relatively atomic, upper continuous lattices. J. Sci. Hiroshima Univ., Ser. A 16, 409–416 (1953).
B. JOnsson: Lattice-theoretic approach to projective and affine geometry. In: Proc. Internat. Sympos. Axiomatic Method, pp. 183–203. Amsterdam North-Holland 1959.
F. Maeda: Perspectivity of points in matroid lattices. J. Sci. Hiroshima Univ., Ser. A-128, 101–112 (1964).
M. F. Janowitz: On the modular relation in atomistic lattices. Fund. Math. (to appear).
R. Wille: Halbkomplementäre Verbände. Math. Z. 94, 1–31 (1966).
F. Maeda: Kontinuierliche Geometrien. Berlin: Springer-Verlag 1958.
R. Baer: Linear algebra and projective geometry. New York: Academic Press 1952.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1970 Springer-Verlag Berlin · Heidelberg
About this chapter
Cite this chapter
Maeda, F., Maeda, S. (1970). Matroid Lattices. In: Theory of Symmetric Lattices. Die Grundlehren der mathematischen Wissenschaften, vol 173. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-46248-1_3
Download citation
DOI: https://doi.org/10.1007/978-3-642-46248-1_3
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-46250-4
Online ISBN: 978-3-642-46248-1
eBook Packages: Springer Book Archive