It can happen that two equivariant sets with atoms are related by a finitely supported bijection, but not by an equivariant bijection.

# Bipartite graphs without bipartite partition

In the equality symmetry, there is a single-orbit graph with the property that 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 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 to , whenever are distinct atoms. That’s it.

# Decidability results based on a WQO in homogeneous atoms

There are several decision problems for certain computation models with atoms that are decidable when atoms admit certain well quasi order (WQO), and undecidable otherwise. We recall the problems and formulate few questions related to WQOs.

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# Cohomology groups of the torus

Čech cohomology is a way of defining cohomology groups (a topological invariant with algebraic structure) on a topological space. We give a rough description of the construction, in a very special case of a torus, and relate it to the construction of the nonstandard alphabets. Continue reading

# Uniform bound on the size of support of a witness

Let be an equivariant relation. We show that under a certain assumption on , for every , if there exists a witness such that holds, then there also exists one with small support. We show an application to extending homomorphisms.

# Orbit finiteness vs. number of supported elements

A set is orbit finite if and only if the number of its elements supported by atoms grows polynomially with .

# Color-preserving automorphisms in Fraisse classes (new version)

Again, a question closely related to a characterization of *standard atoms* (not to be confused with standard alphabets:). The question is a refinement of a question posted in Automorphisms vs color-preserving automorphisms in Fraisse classes, which has been answered negatively (see the recent post The colored open question closed).

# The colored open question closed

A question in my recent post Automorphisms vs color-preserving automorphisms in Fraisse classes has been answered negatively by Mikołaj. The counterexample Fraisse class is the age of the structure , the rational numbers with the ternary ‘cyclic order’ relation defined by:

I am about to reformulate the question slightly, in order to make refutation harder;)

# A stronger Kleene theorem

This post continues the discussion on the Kleene theorem from here. The previous post gave a notion of regular expressions that had the same expressive power as nondeterministic automata. The idea was that a star corresponds to a graph where the set of vertices had one orbit, and where the edges where labelled by simpler regular expressions. In the case without atoms, such a graph is necessarily a self loop, which corresponds to the standard Kleene star. This post is about a stronger requirement: instead of saying that the graph has one orbit of vertices, we say that it has one orbit of edges.

# Automorphisms vs color-preserving automorphisms in Fraisse classes

Here is a question closely related to a characterization of *standard atoms* (not to be confused with standard alphabets:). It asks whether a bound on the number of automorphisms is equivalent to a bound on the numer of color-preserving automorphisms, in every Fraisse class of finite relational structures.