arXiv subjects

Pimpen Vejjajiva

Publications and source records attributed to Pimpen Vejjajiva.

2 recordsLinked to original sources

arXiv2023

The finite sequences and the partitions whose members are finite of a set

In this paper, we investigate relationships between $|\seq(A)|$ and $|\Part_{\fin}(A)|$ in the absence of the Axiom of Choice, where $\seq(A)$ is the set of finite sequences of elements in a set $A$ and $\Part_{\fin}(A)$ is the set of partitions of $A$ whose members are finite. We show that $|\seq(A)|<|\Part_{\fin}(A)|$ if $A$ is Dedekind-infinite and the condition cannot be removed. Moreover, this relationship holds for an arbitrary infinite set $A$ if we restrict $\seq(A)$ to the set of finite sequences with a bounded length.

math.LO↗

arXiv2023

The partitions whose members are finite and the permutations with at most $n$ non-fixed points of a set

We write $S_{\leq n}(A)$ and $\Part_{\fin}(A)$ for the set of permutations with at most $n$ non-fixed points, where $n$ is a natural number, and the set of partitions whose members are finite, respectively, of a set $A$. Among our results, we show, in the Zermelo-Fraenkel set theory, that $|\Part_{\fin}(A)| \nleq |S_{\leq n}(A)|$ for any infinite set $A$ and if $A$ can be linearly ordered, then $|S_{\leq n}(A)| < |\Part_{\fin}(A)|$ while the statement ``$|S_{\leq n}(A)|\leq|\Part_{\fin}(A)|$ for all infinite sets $A$" is not provable for $n\geq 3$.

math.LO↗