representable posets参考文献.pdf

Journal of Applied Logic 16 (2016) 60–71
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Journal of Applied Logic

Representable posets
Rob Egrot
Faculty of ICT, Mahidol University, 999 Phuttamonthon 4 Road, Salaya, Nakhon Pathom 73170,
Thailand
a r t i c l e i n f o a b s t r a c t
Article history: A poset is representable if it can be embedded in a field of sets in such a way
Received 13 January 2016 that existing finite meets and joins become intersections and unions respectively
Accepted 3 March 2016 (we say finite meets and joins are preserved). More generally, for cardinals α and
Available online 10 March 2016 β a poset is said to be (α, β)-representable if an embedding into a field of sets
Keywords: exists that preserves meets of sets smaller than α and joins of sets smaller than β.
≤ ≤
Poset We show using an ultraproduct/ultraroot argument that when 2 α, β ω
Partially ordered set the class of (α, β)-representable posets is elementary, but does not have a finite
Representation axiomatization in the case where either α or β = ω. We also show that the classes
Axiomatization of posets with representations preserving either countable or all meets and joins are
Elementary class pseudoelementary.
© 2016 Elsevier . All rights reserved.
1. Introduction
Every poset P can be thought of as a set of sets ordered by inclusion by considering the embedding
P → ℘(P ) defined by p →{q ∈ P : q ≤ p}. Indeed, this representation has the advantage of preserving
all existing