# Tensor products of semilattices with zero, revisited

@article{Gratzer2000TensorPO,
title={Tensor products of semilattices with zero, revisited},
author={Georg Gratzer and Friedrich Wehrung},
journal={Journal of Pure and Applied Algebra},
year={2000},
volume={147},
pages={273-301}
}
• Published 2000
• Mathematics
• Journal of Pure and Applied Algebra
Let A and B be lattices with zero. The classical tensor product, $A\otimes B$, of A and B as join-semilattices with zero is a join-semilattice with zero; it is, in general, not a lattice. We deﬁne a very natural condition: $A \otimes B$ is capped (that is, every element is a ﬁnite union of pure tensors) under which the tensor product is always a lattice. Let Conc L denote the join-semilattice with zero of compact congruences of a lattice L. Our main result is that the following isomorphism… Expand
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In general, the tensor product, $A\otimes B$, of the lattices A and B with zero is not a lattice (it is only a join-semilattice with zero). If $A \otimes B$ is a capped tensor product, then $AExpand A New Lattice Construction: The Box Product • Mathematics • 1999 In a recent paper, the authors have proved that for lattices A and B with zero, the isomorphism$Conc(A \otimes B)\cong Conc A \otimes Conc B$, holds, provided that the tensor product satisﬁes a veryExpand Tensor products and relation quantales • Mathematics • 2016 A classical tensor product $${A \otimes B}$$A⊗B of complete lattices A and B, consisting of all down-sets in $${A \times B}$$A×B that are join-closed in either coordinate, is isomorphic to theExpand Solutions to five problems on tensor products of lattices and related matters Abstract. The notion of a capped tensor product, introduced by G. Grätzer and the author, provides a convenient framework for the study of tensor products of lattices that makes it possible to extendExpand Lattice Tensor Products i. Coordinatization • Mathematics • 2002 G. Grätzer and F. Wehrung introduced the lattice tensor product, A ⊠ B, of the lattices A and B. One of the most important properties is that for a simple and bounded lattice A, the lattice A ⊠ B isExpand A non-capped tensor product of lattices In lattice theory, the tensor product $${A \otimes B}$$A⊗B is naturally defined on ($${\vee}$$∨, 0)-semilattices. In general, when restricted to lattices, this construction will not yield a lattice.Expand Lattice tensor products. III - Congruences • Mathematics • 2003 G. Grätzer and F. Wehrung introduced the lattice tensor product, A⊠B, of the lattices Aand B. In Part I of this paper, we showed that for any finite lattice A, we can "coordinatize" A⊠B, that is,Expand Flat semilattices • Mathematics • 2005 Let A, B, and S be {∨, 0}-semilattices and let f : A ֒→ B be a {∨, 0}-semilattice embedding. Then the canonical map, f ⊗ id S , of the tensor product A ⊗ S into the tensor product B ⊗ S is notExpand FROM JOIN-IRREDUCIBLES TO DIMENSION THEORY FOR LATTICES WITH CHAIN CONDITIONS For a finite lattice L, the congruence lattice Con L of L can be easily computed from the partially ordered set J(L) of join-irreducible elements of L and the join-dependency relation DL on J(L). WeExpand Tensor products of semilattices and fuzzy ideals • T. Kuraoka • Mathematics, Computer Science • Fuzzy Sets Syst. • 2016 It is shown that the lattice of all complete fuzzy ideals on a semilattice is an extension of tensor product, and the notion of semicomplete bi-ideals is defined. Expand #### References SHOWING 1-10 OF 23 REFERENCES A New Lattice Construction: The Box Product • Mathematics • 1999 In a recent paper, the authors have proved that for lattices A and B with zero, the isomorphism$Conc(A \otimes B)\cong Conc A \otimes Conc B\$, holds, provided that the tensor product satisﬁes a veryExpand
The semilattice tensor product of distributive lattices
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• 1981
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• 1994
Thefunction lattice LP is the lattice of all isotone maps from a posetP into a latticeL.D. Duffus, B. Jónsson, and I. Rival proved in 1978 that for afinite poset P, the congruence lattice ofLP is aExpand
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General Lattice Theory
I First Concepts.- 1 Two Definitions of Lattices.- 2 How to Describe Lattices.- 3 Some Algebraic Concepts.- 4 Polynomials, Identities, and Inequalities.- 5 Free Lattices.- 6 Special Elements.-Expand
Tensor products and transferability
• manuscript
• 1997
Tensor products and transferability, manuscript
• Tensor products and transferability, manuscript
• 1997
Harcourt Brace Jovanovich, Publishers)
• Mathematische Reihe
• 1978