关于正则约化积
On regular reduced products
摘要：
假设<aleph_0，aleph_1>-><lambda，lambda^+>。假设M是一个一阶理论T的模型，其基数在基数<= lambda的词汇表L（T）中至多为lambda^+。设N是一个具有相同词汇表的模型。设Delta是L（T）中的一阶公式集，D是lambda上的正则滤子。则M是Delta-可嵌入到约化幂N^lambda/D中的，条件是每个在M中为真的Delta-存在公式也在N中为真。我们得到如下推论：对M如上所述，D是lambda上的正则超滤子，M^lambda/D是lambda^{++}-泛的。我们的第二个结果如下：对于i<mu，设M_i和N_i是基数<=lambda的词汇表的基本等价模型。假设D是mu上的正则滤子，并且<aleph_0，aleph_1>-><lambda，lambda^+>成立。我们证明了在长度为λ ^+的Escherichfeucht-Fraisse博弈中，第二个局中人在prod_i M_i/D和prod_i N_i/D上有一个获胜策略。这产生了以下推论：假设GCH和lambda正则）。对于L，M_i和N_i，若D是λ上的正则滤子，则prod_i M_i/D cong prod_i N_i/D .
英文摘要：
Assume <aleph_0,aleph_1>-><lambda,lambda^+>. Assume M is a model of a first order theory T of cardinality at most lambda^+ in a vocabulary L(T) of cardinality <= lambda . Let N be a model with the same vocabulary. Let Delta be a set of first order formulas in L(T) and let D be a regular filter on lambda. Then M is Delta-embeddable into the reduced power N^lambda/D, provided that every Delta-existential formula true in M is true also in N. We obtain the following corollary: for M as above and D a regular ultrafilter over lambda, M^lambda/D is lambda^{++}-universal. Our second result is as follows: For i<mu let M_i and N_i be elementarily equivalent models of a vocabulary which has has cardinality <=lambda. Suppose D is a regular filter on mu and <aleph_0,aleph_1>-><lambda,lambda^+> holds. We show that then the second player has a winning strategy in the Ehrenfeucht-Fraisse game of length lambda^+ on prod_i M_i/D and prod_i N_i/D. This yields the following corollary: Assume GCH and lambda regular). For L, M_i and N_i as above, if D is a regular filter on lambda, then prod_i M_i/D cong prod_i N_i/D .
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