Tensor products of semilattices with zero, revisited

Grätzer, George; Wehrung, Friedrich

doi:10.1016/s0022-4049(98)00145-5

Cited by 23 publications

(57 citation statements)

References 12 publications

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“…Quackenbush [7] and G. Grätzer and F. Wehrung [9,10,11]. For finite lattices A and B, it is not always the case that Dim(A ⊗ B) is isomorphic to Dim A ⊗ Dim B, however, we prove that related positive statements hold.…”

Section: Introductionmentioning

confidence: 70%

“…This follows immediately from the representation of S ⊗ T as the lattice of bi-ideals of S × T , see [9].…”

Section: Then There Exists a Unique Monoid Homomorphismmentioning

confidence: 97%

“…For A and B finite, the lattices C that satisfy the conditions of Proposition 7.1 are called sub-tensor products of A and B in [9]. Lemma 7.2.…”

Section: Then There Exists a Unique Monoid Homomorphismmentioning

confidence: 99%

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From Join-Irreducibles to Dimension Theory for Lattices With Chain Conditions

Wehrung

2002

J. Algebra Appl.

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Abstract. 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 D L on J(L). We establish a similar version of this result for the dimension monoid Dim L of L, a natural precursor of Con L. For L join-semidistributive, this result takes the following form: Theorem 1. Let L be a finite join-semidistributive lattice. Then Dim L is isomorphic to the commutative monoid defined by generators ∆(p), for p ∈ J(L), and relationsAs a consequence of this, we obtain the following results:Theorem 2. Let L be a finite join-semidistributive lattice. Then L is a lower bounded homomorphic image of a free lattice iff Dim L is strongly separative, iff it satisfies the axiom (∀x)(2x = x ⇒ x = 0).

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Section: Introductionmentioning

confidence: 70%

“…This follows immediately from the representation of S ⊗ T as the lattice of bi-ideals of S × T , see [9].…”

Section: Then There Exists a Unique Monoid Homomorphismmentioning

confidence: 97%

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From Join-Irreducibles to Dimension Theory for Lattices With Chain Conditions

Wehrung

2002

J. Algebra Appl.

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“…The construction M L is related to the 3 w x w x Ž . classical construction M L of Schmidt 22 for a distributive lattice L , 3 w x which, in turn, is related to tensor products, see Anderson and Kimura 1 , w x w x w x Fraser 6 , Gratzer et al 12 , and our paper 18 . w x The crucial step was taken in our paper 19 , in which we introduced another lattice construction, the box product, that relates to the tensor ² : w x product just as the M L relates to M L .…”

Section: Strong Independence Theorems For General Latticesmentioning

confidence: 87%

The Strong Independence Theorem for Automorphism Groups and Congruence Lattices of Arbitrary Lattices

Grätzer¹,

Wehrung²

2000

Advances in Applied Mathematics

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“…In our paper [10], we recalled in detail the introduction of tensor products of lattices in the seventies. The main result of this field is the isomorphism we proved in [10] for capped tensor products; this generalizes the result of G. Grätzer, H. Lakser, and R. W. Quackenbush [6] for finite lattices.…”

Section: Introductionmentioning

confidence: 99%

A New Lattice Construction: The Box Product

Grätzer¹,

Wehrung²

1999

Journal of Algebra

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In a recent paper, the authors have proved that for lattices A and B with zero, the isomorphism Con c A ⊗ B ∼ = Con c A ⊗ Con c B holds, provided that the tensor product satisfies a very natural condition (of being capped) implying that A ⊗ B is a lattice. In general, A ⊗ B is not a lattice; for instance, we proved that M 3 ⊗ F 3 is not a lattice.In this paper, we introduce a new lattice construction, the box product for arbitrary lattices. The tensor product construction for complete lattices introduced by G. N. Raney in 1960 and by R. Wille in 1985 and the tensor product construction of A. Fraser in 1978 for semilattices bear some formal resemblance to the new construction.For lattices A and B, while their tensor product A ⊗ B (as semilattices) is not always a lattice, the box product, A I B, is always a lattice. Furthermore, the box product and some of its ideals behave like an improved tensor product. For example, if A and B are lattices with unit, then the isomorphism grätzer and wehrungholds. There are analogous results for lattices A and B with zero and for a bounded lattice A and an arbitrary lattice B.A join semilattice S with zero is called 0 -representable, if there exists a lattice L with zero such that Con c L ∼ = S. The above isomorphism results yield the following consequence: The tensor product of two 0 -representable semilattices is 0 -representable.

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Tensor products of semilattices with zero, revisited

Cited by 23 publications

References 12 publications

From Join-Irreducibles to Dimension Theory for Lattices With Chain Conditions

From Join-Irreducibles to Dimension Theory for Lattices With Chain Conditions

The Strong Independence Theorem for Automorphism Groups and Congruence Lattices of Arbitrary Lattices

A New Lattice Construction: The Box Product

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