Abstract
Grant A. Fraser defined the semilattice tensor productA ⊗B of distributive latticesA, B and showed that it is a distributive lattice. He proved that ifA ⊗B is projective then so areA andB, that ifA andB are finite and projective thenA ⊗B is projective, and he gave two infinite projective distributive lattices whose semilattice tensor product is not projective. We extend these results by proving that ifA andB are distributive lattices with more than one element thenA ⊗B is projective if and only if bothA andB are projective and both have a greatest element.
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Lakser, H. The semilattice tensor product of projective distributive lattices. Algebra Universalis 13, 78–81 (1981). https://doi.org/10.1007/BF02483824
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DOI: https://doi.org/10.1007/BF02483824