# Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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## A Galois connection for reduced incidence algebrasHTML articles powered by AMS MathViewer

by Robert L. Davis
Proc. Amer. Math. Soc. 45 (1974), 179-184 Request permission

## 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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