A structure theory for ordered sets

Duffus, Dwight; Rival, Ivan

doi:10.1016/0012-365x(81)90201-6

Cited by 96 publications

(53 citation statements)

References 30 publications

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“…Duffus and Rival [1] studied a certain form of representability of partially ordered sets. The representation under consideration was defined by means of retracts.…”

Section: Introductionmentioning

confidence: 99%

“…The representation under consideration was defined by means of retracts. In [1] it was remarked that "the following important problem remains unsolved: ( * ) Does every partially ordered set have a representation {P i : i ∈ I} such that P i for each i ∈ I is irreducible?" (The detailed definitions of the notions of representation and irreducibility are recalled in Section 2 below.…”

Section: Introductionmentioning

confidence: 99%

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On a representation of monounary algebras

Jakubíková-Studenovská

2005

Czech Math J

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“…Duffus and Rival [1] studied a certain form of representability of partially ordered sets. The representation under consideration was defined by means of retracts.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On a representation of monounary algebras

Jakubíková-Studenovská

2005

Czech Math J

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“…It is known (see [2,14]) that every complete lattice L with at least two elements has a representation (P x : x ∈ L), where each P x ∼ = 2. It is also well-known that for every poset P that is not an antichain, P n retracts to 2 n .…”

Section: 3mentioning

confidence: 99%

Descending Chains and Antichains of the Unary, Linear, and Monotone Subfunction Relations

Lehtonen¹

2006

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Abstract. The C-subfunction relations on the set of operations on a finite base set A defined by function classes C are examined. For certain clones C on A, it is determined whether the partial orders induced by the respective C-subfunction relations have infinite descending chains or infinite antichains. More specifically, we investigate the subfunction relations defined by the clone of all functions on A, the clones of essentially at most unary operations, the clones of linear functions on a finite field, and the clones of monotone functions with respect to the various partial orders on A.

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“…(Note that Hom(Q) for such structures is trivial). Retractions play an important role in the structure theory of graphs and orders [11,17,18]. The computational complexity of these problems has been extensively studied [13,14,16,23,27,31] in an attempt to distinguish tractable cases from NP-complete ones.…”

Section: Introductionmentioning

confidence: 99%

First-order Definable Retraction Problems for Posets and Reflexive Graphs

Dalmau

Krokhin

Larose

2006

Journal of Logic and Computation

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Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. Abstract A retraction from a structure P to its substructure Q is a homomorphism from P onto Q that is the identity on Q. We present an algebraic condition which completely characterises all posets and all reflexive graphs Q with the following property: the class of all posets or reflexive graphs, respectively, that admit a retraction onto Q is first-order definable.

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A structure theory for ordered sets

Cited by 96 publications

References 30 publications

On a representation of monounary algebras

On a representation of monounary algebras

Descending Chains and Antichains of the Unary, Linear, and Monotone Subfunction Relations

First-order Definable Retraction Problems for Posets and Reflexive Graphs

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