Η set

In mathematics, an η set (eta set) is a type of totally ordered set introduced by that generalizes the order type η of the rational numbers.

Definition

If is an ordinal then an set is a totally ordered set in which for any two subsets and of cardinality less than , if every element of is less than every element of then there is some element greater than all elements of and less than all elements of .

Examples

The only non-empty countable η₀ set (up to isomorphism) is the ordered set of rational numbers.

Suppose that κ = Ã¢Â„µ_α is a regular cardinal and let X be the set of all functions f from κ to {−1,0,1} such that if f(α) = 0 then f(β) = 0 for all β > ÃŽÂ±, ordered lexicographically. Then X is a η_α set. The direct limit of all these orders is isomorphic to the class of surreal numbers.

A dense totally ordered set without endpoints is an η_α set if and only if it is ℵ_α saturated.

Properties

Any η_α set X is universal for totally ordered sets of cardinality at most ℵ_α, meaning that any such set can be embedded into X.

For any given ordinal α, any two η_α sets of cardinality ℵ_α are isomorphic (as ordered sets). An η_α set of cardinality ℵ_α exists if ℵ_α is regular and Σ_{β<α} 2^{ℵ_β}&nbsp;≤&nbsp;ℵ_α.

References

• English translation in