Tag Archives: equality atoms

Automorphism groups of definable structures

We consider equality atoms \atoms, i.e., \atoms is a countable set with the equality relation. Recall that by definable set we mean a structure which can be defined (in a possibly nested fashion) from \atoms using finite unions, finite tuples, and set-builder expressions with first-order formulas ranging over \atoms, e.g. \set{(x,y):x,y\in\atoms, x\neq y}\cup\set{x:x\in\atoms, x\neq 5} is definable using the parameter 5. A definable relational structure is a relational structure (A,R_1,\ldots,R_n) where A and each relation R_i is a definable set. Up to isomorphism, definable structures are the same as structures which interpret in \atoms, using first-order interpretations (this correspondence preserves parameters), but sometimes using definable sets makes constructions easier.

The problem described in the previous post is to determine isomorphism between two definable structures. In this post, we describe this problem in terms of automorphism groups. This post is based on discussions with Manuel Bodirsky and Antoine Mottet.

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Graph isomorphism

Fix an \omega-categorical structure \atoms. The simplest, and already interesting case is (\mathbb N,=), the pure countable set.

We consider the definable graph isomorphism problem for graphs which are definable (equivalently, interpret in) \atoms:

Problem: DefGraphIso

Input: Two definable graphs G,H

Output: Yes, if G and H are isomorphic, No otherwise.

Below we make some very basic observations concerning this problem.

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Can orbit-finite semigroups be straightened?

A straight set is an equivariant set which is equivariantly isomorphic to a disjiont union of sets of the form \atoms^{(n)}. A straight semigroup is an equivariant semigroup whose universe is a straight set. We raise the following:

Question 1: is every equivariant, orbit-finite semigroup an image of some orbit-finite, straight semigroup under an equivariant mapping?

We show some simple preliminary observations towards the above question.

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Orbit-finite bijections can be smoothed

In his recent post Bartek demonstrated an example of two equivariant sets with atoms that are related by a finitely supported bijection, but not by any equivariant bijection. We show that if the sets are orbit-finite then this situation is impossible.

Proposition. Let X and Y be two orbit-finite equivariant sets with atoms. If there exists a finitely supported bijection f \colon X \rightarrow Y then there exists an equivariant bijection F \colon X \rightarrow Y.
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Bipartite graphs without bipartite partition

In the equality symmetry, there is a single-orbit graph G=(V,E) with the property that G is bipartite in the classical sense (equivalently, has no odd-length cycle) but has no partition into two parts which are finitely supported and not inter-connected. In other words, the graph G maps to the single-edge graph via a infinitely-supported homomorphism, but not via a finitely-supported one. And here’s the graph: its nodes are pairs of distinct atoms, and there is an edge from (a,b) to (b,a), whenever a,b are distinct atoms. That’s it.