nLab
continuous poset

Continuous posets

Definitions

Let PP be a poset with directed joins.

Elementary definition

Definition

For a,bPa, b \in P, we say that aa is way below bb and write aba \ll b if whenever SPS \subseteq P is a directed subset and bSb \leq \bigvee S (where S\bigvee S denotes the join of SS), then there exists sSs \in S with asa \leq s.

Definition

We say that PP is continuous if for every aPa\in P, the subset

(a){bP|ba} \Downarrow (a) \coloneqq \{ b \in P | b \ll a \}

is directed and has join aa.

If PP also has finite joins, hence is a suplattice, then (a)\Downarrow(a) is automatically directed, so the condition reduces to ((a))=a\bigvee (\Downarrow(a)) = a; in this case PP is called a continuous lattice.

Adjoint characterization

Let Idl(P)Idl(P) denote the poset of ideals in PP, i.e. subsets that are downward-closed and upwards-directed.

Proposition

A poset PP has directed joins if and only if the principal-ideal map :PIdl(P)\downarrow : P \to Idl(P), defined by

(a){bP|ba}, \downarrow(a) \coloneqq \{ b \in P | b \leq a \} ,

has a left adjoint, which must be \bigvee.

Theorem

A poset PP with directed joins is continuous if and only if :Idl(P)P\bigvee: Idl(P) \to P has its own left adjoint, which must be \Downarrow.

This characterization generalizes directly to the notion of continuous category.

Morphisms

Since a continuous poset must have directed joins, the obvious morphisms in a category whose objects are continuous posets would be the Scott-continuous functions, that is those that preserve directed joins. (These always preserve the order; that is, they are monotone functions.)

Between continuous lattices, we may use the same morphisms; or we may more generally use those Scott-continuous functions that preserve the semilattice structure of finitary joins, in other words, the suplattice morphisms that preserve all joins. However, because a suplattice is a complete lattice, another common choice is to use the Scott-continuous functions that are also inflattice morphisms, that is those that also preserve all meets. Another choice might be the complete-lattice morphisms, those that preserve all meets and all joins.

Properties

References

  • R.-E. Hoffmann, Continuous posets and adjoint sequences , Semigroup Forum 18 (1979) pp.173-188. (gdz)

Last revised on October 27, 2016 at 07:49:00. See the history of this page for a list of all contributions to it.