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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

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: If $ A$ and $ B$ are finite lattices, then the tensor product $ C$ of $ A$ and $ B$ 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 $ C$ as the free product (in the category of bounded distributive lattices) of the congruence lattice of $ A$ and the congruence lattice of $ B$. This provides us with a method of constructing finite subdirectly irreducible (resp., simple) lattices: if $ A$ and $ B$ 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 $ {M_3}$.


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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