The structure of tensor products of semilattices with zero

Authors:
G. Grätzer, H. Lakser and R. Quackenbush

Journal:
Trans. Amer. Math. Soc. **267** (1981), 503-515

MSC:
Primary 06B05

DOI:
https://doi.org/10.1090/S0002-9947-1981-0626486-8

MathSciNet review:
626486

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Abstract | References | Similar Articles | Additional Information

Abstract: If and are finite lattices, then the tensor product of and in the category of join semilattices with zero is a lattice again. The main result of this paper is the description of the congruence lattice of as the free product (in the category of bounded distributive lattices) of the congruence lattice of and the congruence lattice of . This provides us with a method of constructing finite subdirectly irreducible (resp., simple) lattices: if and are finite subdirectly irreducible (resp., simple) lattices then so is their tensor product. Another application is a result of E. T. Schmidt describing the congruence lattice of a bounded distributive extension of .

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1981-0626486-8

Keywords:
Semilattice,
lattice,
tensor product,
congruence lattice,
simple,
subdirectly irreducible

Article copyright:
© Copyright 1981
American Mathematical Society