# Power-set in Hypercube: historical background of indexing each term like Hasse Diagram?

My instructor wants references about the indexation over the hypercube, related question here, he does not believe that I was the first who used it -- [update] thanks to a comment, the name is Hasse Diagram for posets. I have seen similar indexation in set-theory related to power-sets like the below. So

what is the historical background about naming vertices in a hypercube?

Example with the 3th Hypercube and the power-set of {1,2,3}

Helper questions

1. Is there someone who worked with power-sets and the axiom of power-set or Zermelo–Fraenkel set theory that used similar notation?

2. Is this somehow related to traversing a hypercube and anti-chains in a poset?

• I don't know how this is even remotely a set theoretical thing. Maybe graph theoretical, or group theoretical? – Asaf Karagila Nov 6 '13 at 19:47
• Isn't this just a consequence of the fact that there are $2^n$ subsets of an $n$-element set, for example the set $\{1,\,2,\,3,\,\dots , \, n\},$ and there are $2^n$ vertices of an $n$-dimensional cube? – Dave L. Renfro Nov 6 '13 at 22:09
• In addition to my last comment, perhaps a version of "Pohlke's Theorem" for hypercubes can be applied, if such a result exists. I don't know anything about this topic, but googling the following three words (not phrase search) seems to bring up some possibly relevant things: "Mehrdimensionale" "Axonometrie" "hypercube" – Dave L. Renfro Nov 6 '13 at 22:32
• This seems to be equivalent to the Hasse diagram of the power set of $\{1,2,\dots,n\}$, with the partial order given by inclusion. The Wikipedia article has some historical references. – Per Manne Nov 7 '13 at 15:36
• This is isomorphic (as a poset or a lattice) to a boolean algebra. IIRC you're in a technical university. Surely you have mates who know this from C-programming: the lattice of $n$-cube is that of the bitmasks of $n$ bits. The join and meet correspond to bitwise OR and AND. But diagrams like this surely predate computers by a comfortable margin. – Jyrki Lahtonen Nov 19 '13 at 7:45