(Or “smash sum”) In domain theory, the coalesced sum of domains A and B, A (+) B, contains all the non-bottom elements of both domains, tagged to show which part 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 a single bottom in the sum.
This may be generalised to any number 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 “(+)” as LaTeX \oplus – a “+” in a circle.
