2016
DOI: 10.1016/j.jal.2016.03.003
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Representable posets

Abstract: A poset is representable if it can be embedded in a field of sets in such a way that existing finite meets and joins become intersections and unions respectively (we say finite meets and joins are preserved). More generally, for cardinals α and β a poset is said to be (α, β)-representable if an embedding into a field of sets exists that preserves meets of sets smaller than α and joins of sets smaller than β. We show using an ultraproduct/ultraroot argument that when 2 ≤ α, β ≤ ω the class of (α, β)-representab… Show more

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Cited by 7 publications
(17 citation statements)
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“…Proof. If x ≤ y, then either: (1) x ∈ N 0 and y = p, (2) x ∈ N n and y ∈ N n , (3) x ∈ N n and y ∈ N n−1 , (4) x = q and y ∈ N k , or (5) x = y. We note that p has no upper bound other than itself, and that this is also true for elements of N n for all 0 ≤ n ≤ k. Proof.…”
Section: No Other Elements Are Comparablementioning
confidence: 99%
See 1 more Smart Citation
“…Proof. If x ≤ y, then either: (1) x ∈ N 0 and y = p, (2) x ∈ N n and y ∈ N n , (3) x ∈ N n and y ∈ N n−1 , (4) x = q and y ∈ N k , or (5) x = y. We note that p has no upper bound other than itself, and that this is also true for elements of N n for all 0 ≤ n ≤ k. Proof.…”
Section: No Other Elements Are Comparablementioning
confidence: 99%
“…In Section 2 we introduce the basic notation, definitions and results for representable posets (using the notation of [3]). Finally in Section 3 we construct the required sequence of posets and prove the necessary results to support our main claim.…”
Section: Introductionmentioning
confidence: 99%
“…A partially ordered set (poset) is representable if it can be embedded into a powerset algebra via a map that preserves existing finite meets and joins. The class of representable posets and its infinitary variations have been studied, not always using this terminology, in [8, 1115, 21, 29, 39], generalising work done in the setting of semilattices [2, 9, 27, 33], and for distributive lattices and Boolean algebras [1, 3, 4, 6, 7, 16, 31, 35, 36]. At first glance, it is far from obvious that the class of representable posets is elementary.…”
Section: Introductionmentioning
confidence: 99%
“…Now, given the description of the class of representable posets in terms of this ‘separation property’, it is possible to show that it can in fact be axiomatised in first-order logic. [11, Theorem 4.5] does this by proving closure under taking isomorphisms, ultraproducts, and ultraroots and appealing to the Keisler–Shelah theorem [28, 34], and similar can be done by proving closure under taking ultraproducts and elementary substructures and appealing to [20, Theorem 2.13]. Such a non-constructive proof of existence may be regarded as being of limited practical use, however, the very fact that an axiomatisation is known to exist can be used in a neat trick to show that a certain constructively generated axiomatisation is correct.…”
Section: Introductionmentioning
confidence: 99%
“…The property of being embeddable into a powerset algebra via an embedding preserving meets and joins of certain cardinalities has been studied using the terminology representable [3,7,14] (see Definition 4.1). A notable departure from the semilattice case is that the first-order theory of representable posets is considerably more complex.…”
Section: Introductionmentioning
confidence: 99%