A trichotomy for a class of equivalence relations

作者：Ding LongYun^*

来源：Science China Mathematics, 2012, 55(12): 2621-2630.

DOI：10.1007/s11425-012-4433-8

• Borel reducibility
• equivalence relation
• metrization

Let X (n) , n is an element of N be a sequence of non-empty sets, psi(n) : X-n(2) -> R+. We consider the relation E = E((X-n , psi(n) )(n is an element of N)) on Pi(n is an element of N) X-n by (x, y) is an element of E((X-n , psi(n)) (n is an element of N)) double left right arrow Sigma(n is an element of N) psi(n) (x(n), y(n)) < + infinity. If E is an equivalence relation and all psi(n), n is an element of N, are Borel, we show a trichotomy that either R-N/l(1) <= (B) E, E-1 <= (B) E, or E <=(B) E-0.
We also prove that, for a rather general case, E((X-n , psi(n) )(n is an element of N)) is an equivalence relation iff it is an l(p)-like equivalence relation.

• 出版日期2012-12
• 单位南开大学