Mark Gritter (markgritter) wrote,
Mark Gritter

gcd's of infinities

A standard construction, courtesty of Wikipedia:

The non-negative integers, ordered by divisibility [form a Complete Lattice]. The least element of this lattice is the number 1, since it divides any other number. Maybe surprisingly, the greatest element is 0, because it can be divided by any other number. The supremum [join] of finite sets is given by the least common multiple and the infimum [meet] by the greatest common divisor. For infinite sets, the supremum will always be 0 while the infimum can well be greater than 1. For example, the set of all even numbers has 2 as the greatest common divisor. If 0 is removed from this structure it remains a lattice but ceases to be complete.

What if we wanted to take this in a nonstandard direction instead? We could imagine some sort of transfinite number that was divisible by all powers of two but not by any other number, call it Ω2. The join of the (infinite) set of numbers divisible by three is then Ω3, and join( Ω2, Ω3 ) = Ω6, naturally. I don't think these behave like the surreal numbers 2ω, 3ω, etc., because I think 3 divides the surreal 2ω = ω + ω but not the "2^infinity" Ω2. (But I'd have to do some more work to be sure.) I would also need to do research to see whether 2ω, 3ω are well defined and fit the bill--- exponentiation is tricky.

It seems like there are two possibilities. Either somebody has already invented this structure and given it a name, or else the idea contains some contradiction. But I don't yet know enough to determine which is the case. :)

Tangentially related: Transfinite Chomp, but there the lattice looks much different.
Tags: geek, mathematics
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