A Galois connection for reduced incidence algebras
Author:
Robert L. Davis
Journal:
Proc. Amer. Math. Soc. 45 (1974), 179184
MSC:
Primary 05B20
MathSciNet review:
0363946
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Abstract: If , and is an equivalence relation on the ``entries'' of the reduced incidence space is the set of all real matrices with support in and such that whenever . Let be the lattice of all subspaces of having support contained in , and that of all equivalences on . Then the map defined above is Galois connected with a map which sends a subspace into the equivalence having whenever all in have . The Galois closed subspaces (i.e. reduced incidence spaces) are shown to be just those subspaces which are closed under Hadamard multiplication, and if is also a subalgebra then its support must be a transitive relation. Consequences include not only pinpointing the role of Hadamard multiplication in characterizing reduced incidence algebras, but methods for constructing interesting new types of algebras of matrices.
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 Garrett Birkhoff, Lattice theory, 3rd ed., Amer. Math. Soc. Colloq. Publ., vol. 25, Amer. Math. Soc., Providence, R. I., 1967. MR 37 #2638. MR 0227053 (37:2638)
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 Peter Doubilet, G.C. Rota and R. P. Stanley, On the foundations of combinatorial theory. VI. The idea of generating function, Proc. Sixth Berkeley Sympos. Math. Statist. and Probability, vol. 2, Univ. of California, Los Angeles, Calif., 1972, pp. 267318. MR 0403987 (53:7796)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939197403639463
PII:
S 00029939(1974)03639463
Keywords:
Reduced incidence algebra,
incidence algebra,
Galois connection,
Hadamard product
Article copyright:
© Copyright 1974
American Mathematical Society
