# What Happened to “Top” in Denotational Semantics?

Has "top" ($$\top$$) been removed from or relegated to a much more minor role in denotational semantics? If so, when and why?

I see older papers and books talking about both "top" and "bottom" ("bot", $$\bot$$), but it seems like newer work only refers to bot. I vaguely recall seeing a reference somewhere that said that top wasn't used anymore or wasn't necessary, but I don't know where that was and I don't know anything about the transition away.

I'm reading a paper from 1979 about applying denotational semantics to databases and null values ("Null Values in Data Base Management: a Denotational Semantics Approach" by Yannis Vassiliou) that makes heavy use of both bot and top, and I'd like to understand what a more modern take on that might be. Any relevant references on that angle would also be greatly appreciated.

• I don't know about the claim. Is it possible that the recent papers you have been reading have been using a partial order while the older papers were using a lattice. Is it possible that the difference was not old vs new but rather whether they needed/used a partial order vs a lattice? To learn more about partial orders in computer science, I recommend matt.might.net/articles/partial-orders. – D.W. Sep 22 '18 at 6:39
• @D.W. Yes. As I recall, all of the relatively recent places I've examined have been using a partial order and not a lattice. Clearly my impressions are subjective and are not comprehensive, but I can't think of a reference to top in these relatively recent examples, and they never seem to require a lattice, instead going for a partial order. On the other hand, the impression I get from materials from roughly the late 70's is that top and lattices are not optional. Is my impression flawed? Are there recent papers that make essential use of top? Is it just a choice tailored to the subject? – joeA Sep 22 '18 at 14:11
• I don't think you're going to find anyone who says that a lattice is not optional / a lattice is optional. These are just tools that you can use or not, and different tools are appropriate for different purposes. – D.W. Sep 22 '18 at 15:26