A Galois Connection for Reduced Incidence Algebras

If (V = j 1, • • • , n |, D CN x 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.. = a whenrr ij rs ever (i, j)F(r, s). Let S(D) be the lattice of all subspaces of R having support contained in D, and S(D) that of all equivalences on D. Then the map g defined above is Galois connected with a map / which sends a subspace S into the equivalence f[S) having (¿, /)l/\S)j (r, s) whenever all A in S have a..-a . The Galois closed subspaces (i.e. reduced incidence spaces) ij rs 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. In Doubilet, Rota and Stanley's study [3] (hereafter referred to as D-R-S) of incidence algebras and how they lead to a coherent theory of generating functions the "main working tool" is the notion of reduced incidence algebra. One of the authors' theorems characterized such reduced incidence algebras, among certain subalgebras of the incidence algebra l(P) of a partial order P, in terms of a pointwise multiplication which seems to have no other use in the theory. When the partial order is finite the incidence algebra can be taken to be an algebra of matrices under ordinary matrix multiplication, while pointwise multiplication is usually called Hadamard multiplication. This note elucidates the somewhat surprising role of Hadamard multiplication by considering arbitrary subspaces of the vector space of all real nx n matrices. Those which are in a natural way "reduced incidence spaces" are just those which are Galois closed in a simple Galois correspondence. Transferring the D-R-S Received by the editors July 17, 1972 and, in revised form, August 11, 1973. AMS (MOS) subject classifications (1970). Primary 06A10; Secondary 06A15, 15A30.