Coalesced sum

<theory> (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.