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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A Galois connection for reduced incidence algebras


Author: Robert L. Davis
Journal: Proc. Amer. Math. Soc. 45 (1974), 179-184
MSC: Primary 05B20
MathSciNet review: 0363946
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Abstract: If $ N = \{ 1, \cdots ,n\} ,D \subset N \times N$, and $ F$ is an equivalence relation on the ``entries'' of $ D$ the reduced incidence space $ g(F)$ is the set of all real matrices $ A$ with support in $ D$ and such that $ {a_{ij}} = {a_{rs}}$ whenever $ (i,j)F(r,s)$. Let $ \mathcal{L}(D)$ be the lattice of all subspaces of $ {R_n}$ having support contained in $ D$, and $ \mathcal{E}(D)$ that of all equivalences on $ D$. Then the map $ g$ defined above is Galois connected with a map $ f$ which sends a subspace $ S$ into the equivalence $ f(S)$ having $ (i,j)[f(S)](r,s)$ whenever all $ A$ in $ S$ have $ {a_{ij}} = {a_{rs}}$. The Galois closed subspaces (i.e. reduced incidence spaces) are shown to be just those subspaces which are closed under Hadamard multiplication, and if $ S = g(F)$ is also a subalgebra then its support $ D$ 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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DOI: http://dx.doi.org/10.1090/S0002-9939-1974-0363946-3
PII: S 0002-9939(1974)0363946-3
Keywords: Reduced incidence algebra, incidence algebra, Galois connection, Hadamard product
Article copyright: © Copyright 1974 American Mathematical Society