# coalesced sum

## 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.
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References in classic literature ?
A Noun is a composite significant sound, not marking time, of which no part is in itself significant: for in double or compound words we do not employ the separate parts as if each were in itself significant.
Having picked up a few of our most familiar colloquial expressions, he scattered them about over his conversation whenever they happened to occur to him, turning them, in his high relish for their sound and his general ignorance of their sense, into compound words and repetitions of his own, and always running them into each other, as if they consisted of one long syllable.

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