UC Irvine

In 1958, Sierpi nski asked whether there exists a linear order $X$ that is isomorphic to its lexicographically ordered cube but is not isomorphic to its square. The corresponding question has been answered positively for many different classes of structures, including groups, rings, graphs, Boolean algebras, and topological spaces of various kinds. However, the main result of this thesis is that the answer to Sierpi nski's question is negative: every linear order $X$ that is isomorphic to its cube is already isomorphic to its square. More generally, if $X$ is isomorphic to any one of its finite powers $X^n$, $n > 1$, it is isomorphic to all of them.

The proof relies on a general representation theorem that characterizes, for a fixed structure $A$ from a class of structures $\mathfrak{C}$, those structures $X \in \mathfrak{C}$ that satisfy the isomorphism $A \times X \cong X$. This characterization is based on an analysis of an arbitrary bijection $f: A\times X \rightarrow X$, and is closely connected to the tail-equivalence relation on the Baire space $A^{\omega}$.

In Chapter 1, we study the tail-equivalence relation as well as those continuous maps on $A^{\omega}$ that preserve tail-equivalence. In Chapter 2, we give our characterization of the isomorphism $A \times X \cong X$, and specify it for several particular classes of structures, including the class of linear orders in which we are primarily interested. In Chapter 3, we use this characterization to solve Sierpi nski's problem, as well as several other problems concerning the multiplication of linear orders. In Chapter 4, we solve a related problem, also due to Sierpi nski, by showing there exist non-isomorphic orders $X$ and $Y$ that divide one another on both sides.