Finite sublattices of a free lattice

Nation, J. B.

doi:10.1090/s0002-9947-1982-0637041-9

Cited by 25 publications

(4 citation statements)

References 22 publications

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“…Nation, to whom this paper is dedicated. Some of these results are included in [7,8,10,11,14,15] and in the monograph Freese, Ježek, and Nation [9], but this list is far from being complete. The monograph just mentioned serves as the reference book for the present paper.…”

Section: Introduction and Our Resultsmentioning

confidence: 99%

Symmetric embeddings of free lattices into each other

2019

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By a 1941 result of Ph. M. Whitman, the free lattice FL(3) on three generators includes a sublattice S that is isomorphic to the lattice FL(ω) = FL(ℵ0) generated freely by denumerably many elements. The first author has recently "symmetrized" this classical result by constructing a sublattice S ∼ = FL(ω) of FL(3) such that S is selfdually positioned in FL(3) in the sense that it is invariant under the natural dual automorphism of FL(3) that keeps each of the three free generators fixed. Now we move to the furthest in terms of symmetry by constructing a selfdually positioned sublattice S ∼ = FL(ω) of FL(3) such that every element of S is fixed by all automorphisms of FL(3). That is, in our terminology, we embed FL(ω) into FL(3) in a totally symmetric way. Our main result determines all pairs (κ, λ) of cardinals greater than 2 such that FL(κ) is embeddable into FL(λ) in a totally symmetric way. Also, we relax the stipulations on S ∼ = FL(κ) by requiring only that S is closed with respect to the automorphisms of FL(λ), or S is selfdually positioned and closed with respect to the automorphisms; we determine the corresponding pairs (κ, λ) even in these two cases. We reaffirm some of our calculations with a computer program developed by the first author. This program is for the word problem of free lattices, it runs under Windows, and it is freely available.

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Section: Introduction and Our Resultsmentioning

confidence: 99%

Symmetric embeddings of free lattices into each other

2019

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“…McKenzie in [31] showed that splitting lattices are useful for investigating non modular varieties of lattices, thus putting the concept in the forefront; a classification of all finite lattices that are splitting for the variety of all lattices followed [32]. In the same period R. McKenzie was proving results about splitting lattices, in another part of the world V. Jankov was studying intermediate logics, equivalently varieties of Heyting algebras; what he did, in our language, was to show that any finite subdirectly irreducible Heyting algebra is splitting for any variety of Heyting algebras to which it belongs [23].…”

Section: Introductionmentioning

confidence: 99%

Splittings in GBL-algebras I: The general case

Aglianò

2019

Fuzzy Sets and Systems

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“…Finite sublattices of free lattices can be characterized by using Whitman's condition and a property involving join covers of elements [8]. Later on, this characterization was strenghened to requiring only the semidistributive laws and Whitman's condition [12]. Regarding properties that are satisfied by all sublattices of free lattices, the following is known.…”

Section: Introductionmentioning

confidence: 99%

Generalizing Galvin and Jónsson's Classification to N5

Chan¹

2019

Preprint

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The problem of determining (up to lattice isomorphism) which lattices are sublattices of free lattices is in general an extremely difficult and an unsolved problem. A notable result towards solving this problem was established by Galvin and Jónsson when they classified (up to lattice isomorphism) all of the distributive sublattices of free lattices in 1959. In this paper, we weaken the requirement that a sublattice of a free lattice be distributive to requiring that a such a lattice belongs in the variety of lattices generated by the pentagon N 5 . Specifically, we use McKenzie's list of join-irreducible covers of the variety generated by N 5 to extend Galvin and Jónsson's results by proving that all sublattices of a free lattice that belong to the variety generated by N 5 satisfy three structural properties. Afterwards, we explain how the results in this paper can be partially extended to lattices from seven known infinite sequences of semidistributive lattice varieties.

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Finite sublattices of a free lattice

Abstract: Abstract. Every finite semidistributive lattice satisfying Whitman's condition is isomorphic to a sublattice of a free lattice.

Cited by 25 publications

References 22 publications

Symmetric embeddings of free lattices into each other

Symmetric embeddings of free lattices into each other

Splittings in GBL-algebras I: The general case

Generalizing Galvin and Jónsson's Classification to N5

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