coalesced sum

coalesced sum

(theory)(Or "smash sum") In domain theory, the coalescedsum of domains A and B, A (+) B, contains all thenon-bottom elements of both domains, tagged to show whichpart of the sum they come from, and a new bottom element.

D (+) E = { bottom(D(+)E) }U { (0,d) | d in D, d /= bottom(D) }U { (1,e) | e in E, e /= bottom(E) }

The bottoms of the constituent domains are coalesced into asingle bottom in the sum. This may be generalised to anynumber of domains.

The ordering is

bottom(D(+)E) <= v For all v in D(+)E

(i,v1) <= (j,v2) iff i = j & v1 <= v2

"<=" is usually written as LaTeX \\sqsubseteq and "(+)" asLaTeX \\oplus - a "+" in a circle.