The semilattice tensor product of distributive lattices

Author:
Grant A. Fraser

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

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Abstract: We define the tensor product 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 is a distributive lattice whenever *A* and *B* are distributive lattices, and we investigate the relationship between the Stone space of 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.

**[1]**R. Balbes,*Projective and injective distributive lattices*, Pacific J. Math.**21**(1967), 405-420. MR**35**#2802. MR**0211927 (35:2802)****[2]**R. Balbes and A. Horn,*Projective distributive lattices*, Pacific J. Math.**33**(1970), 273-279. MR**43**#121. MR**0274356 (43:121)****[3]**G. 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)****[4]**N. Bourbaki,*Algèbre*. Chap. 3:*Algèbre multilinéaire*, Actualités Sci. Indust., no. 1044, Hermann, Paris, 1958. MR**30**#3104.**[5]**G. Grätzer,*Universal algebra*, Van Nostrand, Princeton, N.J., 1968. MR**40**#1320. MR**0248066 (40:1320)****[6]**-,*Lattice theory. First concepts and distributive lattices*, Freeman, San Francisco, Calif., 1971. MR**48**#184. MR**0321817 (48:184)****[7]**W. Greub,*Multilinear algebra*, Springer-Verlag, New York, 1967. MR**37**#222. MR**0224623 (37:222)****[8]**S. Mac Lane,*Homology*, Die Grundlehren der math. Wissenschaften, Band 114, Springer-Verlag, Berlin and New York, 1963. MR**28**#122.**[9]**D. Mowat,*A Galois problem for mappings*, Ph. D. Thesis, University of Waterloo, 1968. MR**0232714 (38:1037)****[10]**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.**[11]**M. H. Stone,*Topological representations of distributive lattices and Brouwerian logics*, Casopis Pěst. Mat. Fys.**67**(1937), 1-25.**[12]**A. Waterman,*Tensor products of lattices*, Harvard University, 1963 (preliminary report).

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1976-0392728-8

Article copyright:
© Copyright 1976
American Mathematical Society