If N = { 1 , ⋯ , n } , D ⊂ N × N N = \{ 1, \cdots ,n\} ,D \subset N \times N , and F F is an equivalence relation on the “entries” of D D the reduced incidence space g ( F ) g(F) is the set of all real matrices A A with support in D D and such that a i j = a r s {a_{ij}} = {a_{rs}} whenever ( i , j ) F ( r , s ) (i,j)F(r,s) . Let L ( D ) \mathcal {L}(D) be the lattice of all subspaces of R n {R_n} having support contained in D D , and E ( D ) \mathcal {E}(D) that of all equivalences on D D . Then the map g g defined above is Galois connected with a map f f which sends a subspace S S into the equivalence f ( S ) f(S) having ( i , j ) [ f ( S ) ] ( r , s ) (i,j)[f(S)](r,s) whenever all A A in S S have a i j = a r s {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 ) S = g(F) is also a subalgebra then its support D 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.