A Galois connection for reduced incidence algebras

Davis, Robert L.

doi:10.1090/S0002-9939-1974-0363946-3

A Galois connection for reduced incidence algebras
HTML articles powered by AMS MathViewer

by Robert L. Davis PDF

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.

References

Garrett Birkhoff, Lattice theory, 3rd ed., American Mathematical Society Colloquium Publications, Vol. XXV, American Mathematical Society, Providence, R.I., 1967. MR 0227053
Robert L. Davis, Algebras defined by patterns of zeros, J. Combinatorial Theory 9 (1970), 257–260. MR 268208
Peter Doubilet, Gian-Carlo Rota, and Richard Stanley, On the foundations of combinatorial theory. VI. The idea of generating function, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971) Univ. California Press, Berkeley, Calif., 1972, pp. 267–318. MR 0403987
David A. Smith, Incidence functions as generalized arithmetic functions. II, Duke Math. J. 36 (1969), 15–30. MR 242757