Consistent Smyth powerdomains

In this paper, we will introduce a new powerdomain called the consistent Smyth powerdomain, which is a free algebra over a continuous dcpo with a partial continuous binary operator that delivers greatest lower bounds (meets) for pairs of elements with an upper bound (consistent pairs) and it thus called consistent meet operator (denoted by ∧↑∧↑). We will show by methods of topology and order theory that the consistent Smyth powerdomain over a continuous dcpo exists and is a continuous dcpo-∧↑∧↑-semilattice. Moreover, if a continuous dcpo is Lawson compact or algebraic, then its consistent Smyth powerdomain is a Lawson compact continuous L  -domain or an algebraic dcpo-∧↑∧↑-semilattice.