On the distributivity and simple connectivity of plane topological lattices

Anderson, Lee

doi:10.1090/s0002-9947-1959-0102575-x

Cited by 18 publications

(2 citation statements)

References 12 publications

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“…In view of the fact that ∆ c (X, cl) contains no copy of M 3 , the 'short-lived question' with which this section began might be revived in the weakened form: Must a modular topological lattice that is homeomorphic to an n-cell be distributive? This is known to be true for n ≤ 3 [2] [13].…”

Section: A Short-lived Variant Of the Above Question And Another Cons...mentioning

confidence: 99%

Simplicial complexes with lattice structures

Bergman

2017

Algebr. Geom. Topol.

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If L is a finite lattice, we show that there is a natural topological lattice structure on the geometric realization of its order complex ∆(L) (definition recalled below). Lattice-theoretically, the resulting object is a subdirect product of copies of L. We note properties of this construction and of some variants, and pose several questions. For M 3 the 5-element nondistributive modular lattice, ∆(M 3 ) is modular, but its underlying topological space does not admit a structure of distributive lattice, answering a question of Walter Taylor.We also describe a construction of "stitching together" a family of lattices along a common chain, and note how ∆(M 3 ) can be regarded as an example of this construction.05E45, 06B30; 06A07, 57Q99 A lattice structure on ∆(L)I came upon the construction studied here from a direction unrelated to the concept of order complex; so I will first motivate it in roughly the way I discovered it, then recall the order complex construction, which turns out to describe the topological structures of these lattices.

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Section: A Short-lived Variant Of the Above Question And Another Cons...mentioning

confidence: 99%

Simplicial complexes with lattice structures

Bergman

2017

Algebr. Geom. Topol.

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“…W. Anderson in [2] showed that locally convex, connected topological lattices are locally connected. With this fact we may modify Theorem 10 of [17] to read Theorem 1.2 (Roy [17]).…”

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Locally convex topological lattices

Stralka¹

1970

Trans. Amer. Math. Soc.

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Abstract. The main theorem of this paper is: Suppose that L is a topological lattice of finite breadth n. Then L can be embedded in a product of n compact chains if and only if L is locally convex and distributive. With this result it is then shown that the concepts of metrizability and separability are equivalent for locally convex, connected, distributive topological lattices of finite breadth.In [8] R. P. Dilworth proved that every distributive lattice of finite breadth n could be embedded (algebraically) in a product of « chains. Since finite breadth and distributivity are hereditary properties this result served to characterize distributive lattices of finite breadth. Dyer and Shields in [9] and Anderson in [4] (also see question 90 of [7]) asked if a result of a similar nature could be obtained for topological lattices, specifically: Can every compact, connected, metric, distributive topological lattice of breadth « be embedded in an «-cell ? This question was answered affirmatively and more generally by Kirby A. Baker and the present author in [6]. For easier reference this result along with several consequences appears as Theorem 1.3 below.The major result of this paper is more nearly the topological analogue to Dilworth's theorem. We characterize the class of those topological lattices which can be embedded in a finite product of compact chains as the class of locally convex, distributive topological lattices of finite breadth. The class of locally convex topological lattices is rather large. For example, it contains all compact topological lattices [15], all locally compact and connected topological lattices [3], and all discrete lattices. This being the case our result contains those of [6] and [8].We also show that the set of separating points of a locally convex, connected topological lattice is very well behaved. This fact together with our main theorem allows us to prove that separability and metrizability are equivalent for locally convex, connected, distributive topological lattices of finite breadth.The author wishes to express his gratitude to F. Burton Jones for his aid in the preparation of this paper.1. Definitions and preliminary results. A topological lattice is a Hausdorff topological space with a pair of continuous maps A,

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Possible classification of finite-dimensional compact Hausdorff topological algebras

Taylor

2018

Don Pigozzi on Abstract Algebraic Logic, Universal Algebra, and Computer Science

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On the distributivity and simple connectivity of plane topological lattices

Cited by 18 publications

References 12 publications

Simplicial complexes with lattice structures

Simplicial complexes with lattice structures

Locally convex topological lattices

Possible classification of finite-dimensional compact Hausdorff topological algebras

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