A Galois connection for reduced incidence algebras

Author:
Robert L. Davis

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

MSC:
Primary 05B20

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

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*, 3rd ed., Amer. Math. Soc. Colloq. Publ., vol. 25, Amer. Math. Soc., Providence, R. I., 1967. MR**37**#2638. MR**0227053 (37:2638)****[2]**R. L. Davis,*Algebras defined by patterns of zeros*, J. Combinatorial Theory**9**(1970), 257-260. MR**42**#3107. MR**0268208 (42:3107)****[3]**Peter Doubilet, G.-C. Rota and R. P. Stanley,*On the foundations of combinatorial theory*. VI.*The idea of generating function*, Proc. Sixth Berkeley Sympos. Math. Statist. and Probability, vol. 2, Univ. of California, Los Angeles, Calif., 1972, pp. 267-318. MR**0403987 (53:7796)****[4]**D. A. Smith,*Incidence functions as generalized arithmetical functions*. II, Duke Math. J.**36**(1969), 15-30. MR**39**#4084. MR**0242757 (39:4084)**

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Additional Information

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