论文标题
超滤器的一致性
Congruence of ultrafilters
论文作者
论文摘要
我们继续研究$ \ hspace {1mm} \ wideTilde {\ mid} \ hspace {1mm} $上的$ {\ mathbb {n}} $ a $ {\ mathbb {n}} $的集合$β{\ mathbb {n}} $，定义为divisibility的划分。这是一个Quasiorder，因此我们将其视为$ = _ \ sim $ - 等价类的订单，其中$ {\ cal f} = _ \ sim {\ cal G} $表示$ {\ cal f} $ \ hspace {1mm} \ widetilde {\ mid} \ hspace {1mm} $ - 划分。在这里，我们介绍了一个新工具：超级滤波器的一致性Modulo的关系。我们首先回忆起Ultrafilters Modulo的一致性，并证明$ = _ \ sim $ - 等价超滤波器不一定具有相同的残基Modulo $ m \ in {\ Mathbb {n}} $。然后，我们将这种关系概括为以自然的方式将超级滤光片一致。之后，使用迭代的非标准扩展，我们引入了更强的关系，相对于超滤器的添加和乘法具有更好的特性。最后，我们还引入了$ \ hspace {1mm} \ widetilde {\ mid} \ hspace {1mm} $的加强，并证明它也与同一致关系相关。
We continue the research of the relation $\hspace{1mm}\widetilde{\mid}\hspace{1mm}$ on the set $β{\mathbb{N}}$ of ultrafilters on ${\mathbb{N}}$, defined as an extension of the divisibility relation. It is a quasiorder, so we see it as an order on the set of $=_\sim$-equivalence classes, where ${\cal F}=_\sim{\cal G}$ means that ${\cal F}$ and ${\cal G}$ are mutually $\hspace{1mm}\widetilde{\mid}\hspace{1mm}$-divisible. Here we introduce a new tool: a relation of congruence modulo an ultrafilter. We first recall the congruence of ultrafilters modulo an integer and show that $=_\sim$-equivalent ultrafilters do not necessarily have the same residue modulo $m\in {\mathbb{N}}$. Then we generalize this relation to congruence modulo an ultrafilter in a natural way. After that, using iterated nonstandard extensions, we introduce a stronger relation, which has nicer properties with respect to addition and multiplication of ultrafilters. Finally, we also introduce a strengthening of $\hspace{1mm}\widetilde{\mid}\hspace{1mm}$ and show that it also behaves well in relation to the congruence relation.
