Distributive Sublattices of a Free Lattice

Galvin, Fred; Jónsson, Bjàrni

doi:10.4153/cjm-1961-022-8

Cited by 45 publications

(52 citation statements)

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“…Using the result of Kostinsky [21] that a finite lattice is projective in the class of all lattices iff it is a sublattice of a free lattice, and the result of Galvin and Jónsson [12] characterizing the finite sublattices of free lattices that are distributive as exactly those that do not contain a doubly reducible element (one that is both a join and meet), gives the following.…”

Section: Theorem 23 If a Finite Lattice P Is Macneille Transferablementioning

confidence: 99%

“…Galvin and Jónsson [12] further characterize all (not just finite) distributive lattices that have no doubly reducible elements. This will be of use in later considerations for us as well.…”

Section: Corollary 24 If a Finite Lattice P Is Macneille Transferabmentioning

confidence: 99%

“…We next provide notation and detail for the linear sums in Theorem 2.5 of Galvin and Jónsson [12]. This theorem will be used to show that there are finite non-projective distributive lattices that are MacNeille transferable for distributive lattices.…”

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MacNeille transferability and stable classes of Heyting algebras

et al. 2018

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Abstract.A lattice P is transferable for a class of lattices K if whenever P can be embedded into the ideal lattice IK of some K ∈ K, then P can be embedded into K. There is a rich theory of transferability for lattices. Here we introduce the analogous notion of MacNeille transferability, replacing the ideal lattice IK with the MacNeille completion K. Basic properties of MacNeille transferability are developed. Particular attention is paid to MacNeille transferability in the class of Heyting algebras where it relates to stable classes of Heyting algebras, and hence to stable intermediate logics.Mathematics Subject Classification. 06D20, 06B23, 06E15.

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Section: Theorem 23 If a Finite Lattice P Is Macneille Transferablementioning

confidence: 99%

Section: Corollary 24 If a Finite Lattice P Is Macneille Transferabmentioning

confidence: 99%

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MacNeille transferability and stable classes of Heyting algebras

et al. 2018

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“…A notable exception to this is the problem of characterizing arbitrary sublattices of a free lattice. Distributive sublattices of a free lattice were described by Galvin and Jónsson [7], and the arguments of [15], based in large part on Kostinsky [16], show that a finitely generated lattice is embeddable in a free lattice iff it satisfies (W) and the generators are contained in D(L) n D'(L). Beyond this little is known except a few necessary conditions (see [15,6]).…”

Section: Figurementioning

confidence: 99%

Finite sublattices of a free lattice

Nation¹

1982

Trans. Amer. Math. Soc.

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“…(Recall that the distributive sublattices of free lattices are at most countable [7].) In the third section we show that for every infinite cardinal k, there is a quotient sublattice (= interval) of FM (k) which is distributive and has cardinality k. The lattice L was originally constructed with this application in mind.…”

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The variety of modular lattices is not generated by its finite members

Freese¹

1979

Trans. Amer. Math. Soc.

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Abstract. This paper proves the result of the title. It shows that there is a five-variable lattice identity which holds in all finite modular lattices but not in all modular lattices. It is also shown that every free distributive lattice can be embedded into a free modular lattice. An example showing that modular lattice epimorphisms need not be onto is given.We prove the result of the title by constructing a simple modular lattice of length six not in the variety generated by all finite modular lattices. This lattice can be generated by five elements and thus the free modular lattice on five generators, FM (5), is not residually finite.Our lattice is constructed using the technique of Hall and Dilworth [9] and is closely related to their third example. Let F and K be countably infinite fields of characteristics p and q, where p and q are distinct primes. Let Lp be the lattice of subspaces of a four-dimensional vector space over F, Lq the lattice of subspaces of a four-dimensional vector space over K. Two-dimensional quotients (i.e. intervals) in both lattices are always isomorphic to Mu (the two-dimensional lattice with « atoms). Thus Lp and Lq may be glued together over a two-dimensional quotient via [9], and this is our lattice. Notice that if F and K were finite fields we could not carry out the above construction since two-dimensional quotients of Lp would have/)" + 1 atoms and those of Lq would have qm + 1, for some n, m > 1. However these numbers are never equal. To some extent the proof is based on this fact. We prove our result by letting / be a homomorphism from a modular lattice M onto our lattice L. A great deal of the structure of L can be pulled back through / into M. We then assume M is residually finite and using von Neumann's theorem arrive at a contradiction similar to the one described above. K. Baker [1] and R. Wille [20] have constructed varieties of modular lattices not generated by their finite members. Using a lattice constructed by E. T. Schmidt [18] the author has shown there is a variety of modular lattices not even generated by its finite dimensional members [6]. However, Schmidt's

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Distributive Sublattices of a Free Lattice

Cited by 45 publications

References 2 publications

MacNeille transferability and stable classes of Heyting algebras

MacNeille transferability and stable classes of Heyting algebras

Finite sublattices of a free lattice

The variety of modular lattices is not generated by its finite members

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