Distributive partially ordered sets

Hickman, Robert; Monro, G. P.

doi:10.4064/fm-120-2-151-166

Cited by 10 publications

(16 citation statements)

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“…The main step in our proof is a result classifying the standard closure operators on ℘(P) whose lattices of closed sets are frames as being precisely those that can be constructed using a certain recursive procedure (Theorem 3.5). This can be viewed as a generalisation of [12,Theorem 2.7]. The proof of the conjecture about neatest representations is then an easy corollary.…”

Section: Introductionmentioning

confidence: 76%

“…Schein's 3-distributivity can be generalised to the concept of α-distributivity for cardinals α (see Definition 3.1). This notion has been studied when α = n < ω [11], when α = ω [4,20] and when α is any regular cardinal [12]. Note that if m, n ≤ ω with m < n, then n-distributivity trivially implies m-distributivity, but the converse is not true [13].…”

Section: Introductionmentioning

confidence: 99%

“…Notions relating to distributivity have also been studied in the more general setting of partially ordered sets (posets) [12,18]. Note that the concepts in [18] and [12] are not equivalent, even for semilattices (that of [18] is a straightforward generalisation of the distributivity of Grätzer and Schmidt [10], while that of [12] is more closely related to the approach of Schein [15]).…”

Section: Introductionmentioning

confidence: 99%

“…The following example is of particular interest to us. For a semilattice, being an ω-distributive poset, as defined in [12], is equivalent to being an ωdistributive semilattice in the sense used here [12,Proposition 2.3], which is, in turn, equivalent to being embeddable into a powerset algebra via a map preserving all existing finite meets and joins [2]. However, this is not true for arbitrary posets, as the ω-distributivity of [12] is strictly stronger than the powerset algebra embedding property [12,…”

Section: Introductionmentioning

confidence: 99%

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Closure Operators, Frames and Neatest Representations

Egrot

2017

Bull. Aust. Math. Soc.

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Given a poset P and a standard closure operator Γ : ℘(P ) → ℘(P ) we give a necessary and sufficient condition for the lattice of Γ-closed sets of ℘(P ) to be a frame in terms of the recursive construction of the Γ-closure of sets. We use this condition to show that given a set U of distinguished joins from P , the lattice of U-ideals of P fails to be a frame if and only if it fails to be σ-distributive, with σ depending on the cardinalities of sets in U. From this we deduce that if a poset has the property that whenever a ∧ (b ∨ c) is defined for a, b, c ∈ P it is necessarily equal to (a ∧ b) ∨ (a ∧ c), then it has an (ω, 3)-representation. This answers a question from the literature.

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Section: Introductionmentioning

confidence: 76%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Closure Operators, Frames and Neatest Representations

Egrot

2017

Bull. Aust. Math. Soc.

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“…Along these lines, SCHEIN [6] proved that a poset in which the iniimum of every two elements exists and which has a special type of distributive property has a representation as a set (Q, E). There are other types of representations given in [3], [4], and…”

Section: Introductionmentioning

confidence: 99%

Representation of Posets

Cheng¹,

Kemp²

1992

Mathematical Logic Qtrly

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In this paper we give new criterions for left distributive posets to have neatest representations. We also illustrate a construction that would embed left distributive posets into representable semilattices. MSC: 06A10, 06A12.Any poset (P, 5) has at least two representations [l] as a poset (Q, S). One of these two representations has the property that the image of the supremum of any (finite or infinite) subset A of P is the union of the images of the elements of A, while the other representation has the property that the image of the infimum of any (finite or infinite) subset B of P is the intersection of the images of the elements of B. In general, we cannot have a representation which has both of the above mentioned properties. However, Smm (71 proved that i f ( P , 5) is a complemented distributive lattice, i.e., a Boolean Algebra, then it has a representation as a set (Q, E). BIRKHOFF (21 has proved a Stone type representation theorem where the lattice (P, 5) is only distributive. Along these lines, SCHEIN [6] proved that a poset in which the iniimum of every two elements exists and which has a special type of distributive property has a representation as a set (Q, E). There are other types of representations given in [3], [4], and[5]. In Section 1 of this paper, we give some other necessary and suficient conditions for posets with this type of distributivity to be representable. In Section 2, we generalize SCHEIN'S result and finally, in Section 3, we illustrate a construction that would embed existing posets into SCHEIN'S type of semilattices by adjoining infimum. With this procedure one could then show the representability of more general posets. 1. Neatest representations of posets D e f i n i t i o n 1. An order isomorphism f from a poset (P, 5) onto a poset (a S ) is called a neatest representation for P iff for every subset {ai: i E S) of P and for every u, b E P (i) /(Aic ai) = nie J ( a i ) , whenever A;. ui exists, (ii) f(o v b) = f ( a ) u f ( b ) , whenever a v b exists D e f i n i t i o n 2 . Let (P, s) be a poset. (P, s) is lefr meet distributive (1.m.d.) iff for every 19 ZUchr. f. math. Logik

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Distributive laws for concept lattices

Erné

1993

Algebra Universalis

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Distributive partially ordered sets

Cited by 10 publications

References 0 publications

Closure Operators, Frames and Neatest Representations

Closure Operators, Frames and Neatest Representations

Representation of Posets

Distributive laws for concept lattices

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