# 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 3 April 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
Tensor products and transferability of semilattices
• Mathematics
• 1999
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
We define the tensor product A 3 B for arbitrary semilattices A and B. The construction is analogous to one used in ring theory (see 4J, [71, 181) and different from one studied by A. Waterman [121,Expand
The structure of tensor products of semilattices with zero
• Mathematics
• 1981
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 theExpand
Congruence lattices of function lattices
• Mathematics
• 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
Tensor products of structures with interpolation
While it is known that the tensor product of two dimension groups is a dimension group, the corresponding problem for interpolation groups has been open for a while. We solve this problem here, byExpand
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