Abstract
Morphisms and weak morphisms extend the concept of strong maps and maps of combinatorial geometry to the class of finite dimensional semimodular lattices. Each lattice which is the image of a semimodular lattice under a morphism is semimodular. In particular, each finite lattice is semimodular if and only if it is the image of a finite distributive lattice under a morphism. Regular and non-singular weak morphisms may be used to characterize modular and distributive lattices. Each morphism gives rise to a geometric closure operator which in turn determines a quotient of a semimodular lattice. A special quotient, the Higgs lift, is constructed and used to show that each morphism decomposes into elementary morphisms, and that each morphism may be factored into an injection and a contraction.
Similar content being viewed by others
References
Birkhoff, G.,Lattice theory. 3rd edition, New York. Amer. Math. Soc. Colloq. Publ. 27, 1967.
Crapo, H. andRota, G.-C.,Combinatorial geometries. M.I.T. Press, Cambridge, Mass., 1970.
Crawley, P., andDilworth, R. P.,Algebraic theory of lattices. Prentice-Hall, Englewood Cliffs, N.J., 1973.
Faigle, U.,Single-element extension and duality of finite semimodular lattices. (in preparation).
Higgs, D. A.,Maps of geometries. J. London Math. Soc.41 (1966), 612–618.
Higgs, D. A.,Strong maps of geometries. J. Combinatorial Theory Ser. A,5 (1968), 185–191.
Ore, O.,Galois connexions. Trans. Amer. Math. Soc.55 (1944), 493–513.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Faigle, U. Über Morphismen halbmodularer Verbände. Aeq. Math. 21, 53–67 (1980). https://doi.org/10.1007/BF02189341
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02189341