A Galois Connection for Reduced Incidence Algebras

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.