The semilattice tensor product of distributive lattices
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- by Grant A. Fraser
- Trans. Amer. Math. Soc. 217 (1976), 183-194
- DOI: https://doi.org/10.1090/S0002-9947-1976-0392728-8
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Abstract:
We define the tensor product $A \otimes B$ for arbitrary semilattices A and B. The construction is analogous to one used in ring theory (see [4], [7], [8]) and different from one studied by A. Waterman [12], D. Mowat [9], and Z. Shmuely [10]. We show that the semilattice $A \otimes B$ is a distributive lattice whenever A and B are distributive lattices, and we investigate the relationship between the Stone space of $A \otimes B$ and the Stone spaces of the factors A and B. We conclude with some results concerning tensor products that are projective in the category of distributive lattices.References
- Raymond Balbes, Projective and injective distributive lattices, Pacific J. Math. 21 (1967), 405–420. MR 211927, DOI 10.2140/pjm.1967.21.405
- Raymond Balbes and Alfred Horn, Projective distributive lattices, Pacific J. Math. 33 (1970), 273–279. MR 274356, DOI 10.2140/pjm.1970.33.273
- Garrett Birkhoff, Lattice theory, 3rd ed., American Mathematical Society Colloquium Publications, Vol. XXV, American Mathematical Society, Providence, R.I., 1967. MR 0227053 N. Bourbaki, Algèbre. Chap. 3: Algèbre multilinéaire, Actualités Sci. Indust., no. 1044, Hermann, Paris, 1958. MR 30 #3104.
- George Grätzer, Universal algebra, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1968. MR 0248066
- George Grätzer, Lattice theory. First concepts and distributive lattices, W. H. Freeman and Co., San Francisco, Calif., 1971. MR 0321817
- W. H. Greub, Multilinear algebra, Die Grundlehren der mathematischen Wissenschaften, Band 136, Springer-Verlag New York, Inc., New York, 1967. MR 0224623 S. Mac Lane, Homology, Die Grundlehren der math. Wissenschaften, Band 114, Springer-Verlag, Berlin and New York, 1963. MR 28 #122.
- David G. Mowat, A Galois problem for mappings, Bull. Amer. Math. Soc. 74 (1968), 1095–1097. MR 232714, DOI 10.1090/S0002-9904-1968-12060-9 Z. Shmuely, Galois connections. I. The construction of Galois connections. II. A “tensor product” of partially ordered sets, D. Sc. Thesis, Technion, Israel Institute of Technology, 1972. M. H. Stone, Topological representations of distributive lattices and Brouwerian logics, Cǎsopis Pěst. Mat. Fys. 67 (1937), 1-25. A. Waterman, Tensor products of lattices, Harvard University, 1963 (preliminary report).
Bibliographic Information
- © Copyright 1976 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 217 (1976), 183-194
- MSC: Primary 06A35
- DOI: https://doi.org/10.1090/S0002-9947-1976-0392728-8
- MathSciNet review: 0392728