# Projection-stable and zero-dimensional domains

## Description

Continuous directed complete partial orders (continuous dcpo's) are ordered algebraic structures which serve as mathematical models for the semantics of programming languages The class of continuous dcpo's is the closure of the class of algebraic dcpo's under images of Scott-continuous projections p: $D \to D$. The paradigm is the Cantor function p: $C \to C,$ which is a Scott-continuous projection on the Cantor set such that im(p) is isomorphic to the unit interval. A dcpo D is called projection-stable iff for all $p \in \lbrack D{\buildrel{\rm pr}\over\longrightarrow} D\rbrack,$ im(p) is algebraic If all order-dense chains in K(D) are degenerate, then an algebraic dcpo D is projection-stable. The converse is not true. If D has a bottom, then the converse is valid. The class of projection-stable dcpo's is closed under arbitrary products Let D be a continuous L-domain (profinite dcpo) with bottom. Then, $\lbrack D{\buildrel{\rm pr}\over\longrightarrow} D\rbrack$ is a continuous L-domain (profinite dcpo) with bottom iff $\lbrack D{\buildrel{\rm pr}\over\longrightarrow} D\rbrack$ is a continuous dcpo iff D is projection-stable Every dI-domain is projection-stable. The dcpo (Proj$(D), \subseteq)$ is a dI-domain, if D is one. The cartesian closed category of dI-domains is closed under the construct Proj(D). A dI-domain D is isomorphic to Proj(D) iff D has a greatest element. In that case, every $p\in$ Proj(D) is of the form $\lambda x.x\cap c$ The canonical inclusion $\eta\sb{D}\: D \to$ Proj(D) preserves all infima and suprema, and it is also $\lambda$-continuous. The assignment $D \mapsto$ Proj(D) is left adjoint to the forgetful functor from dI-domains with top and maps preserving all suprema. A representation of Proj(D) is the set of all Scott-closed sets $C \subseteq D$ which are normal in D We employ Koch's Arc Theorem. Let D be a continuous dcpo such that for all $x \in D$, the dcpo ${\downarrow}x$ is $\lambda$-compact. Then, D is algebraic iff $\lambda$(D) is zero dimensional iff ($D, \leq, \lambda$(D)) does not contain a homeomorphic copy of the unit interval. We classify those continuous dcpo's D, for which ($D, \lambda$(D)) is a Stone space. They are all algebraic dcpo's D such that K(D) has property M. For a bicomplete dcpo with bottom, the space $(D, \lambda(D))$ is connected iff bottom is the only compact element. If D is $\lambda$-compact, every $x \ne \perp\sb{D}$ is connected with $\perp\sb{D}$ by a homeomorphic copy of the unit interval The class of bounded complete algebraic dcpo's with bottom is the maximal first-order axiomatizable class of $\lambda$-compact algebraic dcpo's with bottom, closed under isomorphisms and finite products