[FOM] typo, choice, and more reals

JoeShipman@aol.com JoeShipman at aol.com
Fri Jul 18 23:35:19 EDT 2003

Schanuel's remarkable construction can be applied to any infinite abelian group G.  When G is the integers, "almost homomorphisms modulo almost equality" is naturally isomorphic to the reals; but as long as you define "almost" to mean "the set of exceptional differences is finite" rather than "the set of exceptional differences is bounded", you get well-defined operations of addition (pointwise) and multiplication (composition) which satisfy the usual rules.

So the real numbers are the special case where the infinite abelian group G is as simple as possible.  What do you get if instead G is Z x (Z/2), or Z x Z, or Z x Z x Z, or (Q, +), or (Q-{0}, *), or Q/Z, or (R, +) ?

-- Joe Shipman

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