<p>The only natural condition I can think of is Berry's "I condition" ([1], Sect. 12.3):</p>

<p>(I) each compact element dominates finitely many elements.</p>

<p>The above condition is the defining property of Berry's dI-domains, which are distributive (that's what the "d" stands for) algebraic domains satisfying condition I.  This is a widely known and well studied class of domains, which served as the basis for what is known as "stable" semantics (an attempt to capture the sequential behavior of the $\lambda$-calculus).  Girard's coherence spaces [2] give a particularly simple and useful example of dI-domains.  A coherence space may be seen as the set of cliques of a reflexive graph, ordered by inclusion.  The compact elements are the finite cliques, which obviously satisfy your requirement.</p>

<p>I don't know if generalizations of the I condition have been studied.  A counterexample to your property is given by the ordinal $\omega+2$ ($\omega+1$ is compact but $\omega$ is not), so certainly generalizing "finite" to "well-founded+no infinite antichains" does not work.</p>

<p>[1]  Roberto Amadio and Pierre-Louis Curien.  Domains and Lambda-Calculi.  Cambridge Tracts in Theoretical Computer Science, 1998.</p>

<p>[2] Jean-Yves Girard, Yves Lafont and Paul Taylor.  Proofs and Types.   Cambridge Tracts in Theoretical Computer Science, 1989.</p>

CC BY-SA 3.0
2015-02-13T13:33:18.373Z
2015-02-13T13:33:18.373Z
Damiano Mazza
<p>In a poset $(D, \sqsubseteq)$, a <em>compact</em> element is an element $d \in D$ such that for every directed set $A$ which happens to have a supremum $\bigsqcup A \in D$ with $d \sqsubseteq \bigsqcup A$, it is $d \sqsubseteq a$ for some $a \in A$ (e.g., Definition I-4.1 of <a href="https://zbmath.org/?q=an:1088.06001" rel="noreferrer">[Gierz et al 2003]</a>).</p>

<p><strong>My (elementary) question</strong>. Suppose $d$ is compact and $c \sqsubseteq d$. What are typical extra impositions on the poset, that ensure that $c$ must also be compact?</p>

CC BY-SA 3.0
2015-02-12T13:55:22.727Z
(hmm, now that I think of it, it might have been better to ask this in math.se...)
Basil
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2015-02-13T13:04:41.797Z
I added some more tags, using only "domain-theory" may have limited the number of people viewing the question.
Damiano Mazza
<pl.programming-languages><lambda-calculus><denotational-semantics><domain-theory>
In which posets is the set of compact elements downwards closed?
Basil
# In which posets is the set of compact elements downwards closed?

(Asked by: username_0 on 2015-02-12)

In a poset $(D, \sqsubseteq)$, a *compact* element is an element $d \in D$ such that for every directed set $A$ which happens to have a supremum $\bigsqcup A \in D$ with $d \sqsubseteq \bigsqcup A$, it is $d \sqsubseteq a$ for some $a \in A$ (e.g., Definition I-4.1 of [[Gierz et al 2003]](https://zbmath.org/?q=an:1088.06001)).

**My (elementary) question**. Suppose $d$ is compact and $c \sqsubseteq d$. What are typical extra impositions on the poset, that ensure that $c$ must also be compact?

**Tags:** pl.programming-languages, lambda-calculus, denotational-semantics, domain-theory
> **Comments:**
> **username_0**: (hmm, now that I think of it, it might have been better to ask this in math.se...)  
> **username_1**: I added some more tags, using only "domain-theory" may have limited the number of people viewing the question.  

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## 1 answers

### Answer by username_1 on 2015-02-13. Score: 7 | Accepted answer

The only natural condition I can think of is Berry's "I condition" ([1], Sect. 12.3):

(I) each compact element dominates finitely many elements.

The above condition is the defining property of Berry's dI-domains, which are distributive (that's what the "d" stands for) algebraic domains satisfying condition I. This is a widely known and well studied class of domains, which served as the basis for what is known as "stable" semantics (an attempt to capture the sequential behavior of the $\lambda$-calculus). Girard's coherence spaces [2] give a particularly simple and useful example of dI-domains. A coherence space may be seen as the set of cliques of a reflexive graph, ordered by inclusion. The compact elements are the finite cliques, which obviously satisfy your requirement.

I don't know if generalizations of the I condition have been studied. A counterexample to your property is given by the ordinal $\omega+2$ ($\omega+1$ is compact but $\omega$ is not), so certainly generalizing "finite" to "well-founded+no infinite antichains" does not work.

[1] Roberto Amadio and Pierre-Louis Curien. Domains and Lambda-Calculi. Cambridge Tracts in Theoretical Computer Science, 1998.

[2] Jean-Yves Girard, Yves Lafont and Paul Taylor. Proofs and Types. Cambridge Tracts in Theoretical Computer Science, 1989.


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