On the number of automorphism of uncontable models

测度代数的自同构群与自然数无限置换群相关的若干群的比较

Comparing the automorphism group of the measure algebra with some groups related to the infinite permutation group of the natural numbers

该数学研究通过直接构造，证明测度代数的自同构群及其保测子群，均无法与P(N)/fin的平凡自同构群同构。

Sets of real numbers closed under Turing equivalence: Applications to fields, orders and automorphisms

In the first half of this paper, we study the way that sets of real numbers closed under Turing equivalence sit inside the real line from the perspective of algebra, measure and order. Afterwards, we combine the results from our study of these sets as orders with a classical construction from Avraham to obtain a restriction about how non trivial automorphism of the Turing degrees (if they exist) interact with $1$-generic degrees.

The structure of random automorphisms of the rational numbers

In order to understand the structure of the "typical" element of an automorphism group, one has to study how large the conjugacy classes of the group are. For the case when typical is meant in the sense of Baire category, Truss proved that there is a co-meagre conjugacy class in Aut(Q, <), the automorphism group of the rational numbers. Following Dougherty and Mycielski we investigate the measure theoretic dual of this problem, using Christensen's notion of Haar null sets. We give a complete description of the size of the conjugacy classes of Aut(Q, <) with respect to this notion. In particula…

The conjugacy problem for automorphism groups of countable homogeneous structures

We consider the conjugacy problem for the automorphism groups of a number of countable homogeneous structures. In each case we find the precise complexity of the conjugacy relation in the sense of Borel reducibility.

surreal 数的自同构群分解

Decomposing the automorphism group of the surreal numbers

本文研究surreal数域的自同构群，通过分解成五个因素来揭示其结构，利用Conway正常形式和Hahn场理论，提供自同构的构造方法，并扩展到强线性自同构和指数结构保持自同构。

Automorphism groups of linearly ordered structures and endomorphisms of the ordered set $(\mathbb{Q},{\leq})$ of rational numbers

We investigate the structure of the monoid of endomorphisms of the ordered set $(\mathbb{Q},{\leq})$ of rational numbers. We show that for any countable linearly ordered set $Ω$, there are uncountably many maximal subgroups of $\operatorname{End}(\mathbb{Q},{\leq})$ isomorphic to the automorphism group of $Ω$. We characterise those subsets $X$ of $\mathbb{Q}$ that arise as a retract in $(\mathbb{Q},{\leq})$ in terms of topological information concerning $X$. Finally, we establish that a countable group arises as the automorphism group of a countable linearly ordered set, and hence as a maximal…