值域有界的一致反对称函数

关于一致反对称函数

On uniformly antisymmetric functions

证明了如果A的子集R是可数的，则总有一致反对称f：A-> {0，1}.我们证明了连续统假设等价于存在一个f：R-> omega，其中对R中的每个x，|S_x| <= 1.如果连续统至少是aleph_n，则存在点x使得S_x至少有2^n-1个元素。我们还证明了存在函数f：Q-> {0，1，2，3}使得S_x总是有限的，但不存在这样的函数在R上具有有限值域

Uniformly bounded arrays and mutually algebraic structures

We define an easily verifiable notion of an atomic formula having uniformly bounded arrays in a structure $M$. We prove that if $T$ is a complete $L$-theory, then $T$ is mutually algebraic if and only if there is some model $M$ of $T$ for which every atomic formula has uniformly bounded arrays. Moreover, an incomplete theory $T$ is mutually algebraic if and only if every atomic formula has uniformly bounded arrays in every model $M$ of $T$.

Functions definable in definably complete uniformly locally o-minimal structure of the second kind

We investigate continuous functions definable in a definably complete uniformly locally o-minimal expansion of the second kind of a densely linearly ordered abelian group (DCULOAS structure). We prove a variant of the Arzela-Ascoli theorem for uniformly continuous definable functions and the following assertion: Consider the parameterized function $f:C \times P \rightarrow M$ which is equi-continuous with respect to $P$. The projection image of the set at which $f$ is discontinuous to the parameter space $P$ is of dimension smaller than $\dim P$ when $C$ is closed and bounded. In addition, we …