A Galois connection for reduced incidence algebras

Author:
Robert L. Davis

Journal:
Proc. Amer. Math. Soc. **45** (1974), 179-184

MSC:
Primary 05B20

MathSciNet review:
0363946

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Abstract: If , and is an equivalence relation on the ``entries'' of the reduced incidence space is the set of all real matrices with support in and such that whenever . Let be the lattice of all subspaces of having support contained in , and that of all equivalences on . Then the map defined above is Galois connected with a map which sends a subspace into the equivalence having whenever all in have . The Galois closed subspaces (i.e. reduced incidence spaces) are shown to be just those subspaces which are closed under Hadamard multiplication, and if is also a subalgebra then its support 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.

**[1]**Garrett Birkhoff,*Lattice theory*, Third edition. American Mathematical Society Colloquium Publications, Vol. XXV, American Mathematical Society, Providence, R.I., 1967. MR**0227053****[2]**Robert L. Davis,*Algebras defined by patterns of zeros*, J. Combinatorial Theory**9**(1970), 257–260. MR**0268208****[3]**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****[4]**David A. Smith,*Incidence functions as generalized arithmetic functions. II*, Duke Math. J.**36**(1969), 15–30. MR**0242757**

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DOI:
https://doi.org/10.1090/S0002-9939-1974-0363946-3

Keywords:
Reduced incidence algebra,
incidence algebra,
Galois connection,
Hadamard product

Article copyright:
© Copyright 1974
American Mathematical Society