Category Archives: posts

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 CSPs

Consider the following decision problem (for classical Turing machines). Let \Sigma be a fixed, finite signature.

Input: Two orbit-finite relational structures \str A,\str B, over \Sigma.

Decide: is there a homomorphism from \str A to \str B?

Question 1. Is the above problem decidable?

Observe that we are not asking for the existence of an equivariant homomorphism. If this were the case, the problem would be clearly decidable, since there are only finitely many equivariant homomorphisms between orbit-finite sets, and they can be effectively scanned. In this problem, the homomorphisms do not even have to be finitely supported! Below is an example where a homomorphism exists, but no finitely supported homomorphism exists.

Example. Consider the equality atoms. Let \str A be the graph whose vertices are pairs of distinct atoms, and in which edges join a pair with its inverse. Let \str B be the 2-clique. Then \str A maps to \str B, but no finitely supported homomorphism exists.

Following the idea of this post, in order to solve this problem, it might be useful to move to the totally ordered atoms (\str Q,\le).

Question 2. In the totally ordered atoms, if there is a homomorphism between orbit-finite structures \str A, \str B, is there one which is equivariant?

A positive answer to Question 2 would imply a positive answer to Question 1.

Edit. After reading Bartek’s comment below, I noticed that Question 2 has an even more negative answer. Consider the structures \str A=(\str Q,<) and \str B=(\str Q,>) (orders reversed). There is a homomorphism from \str A to \str B, namely the mapping x\mapsto -x, but there is none which is finitely supported.


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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Join of two elements

Let a,b be two elements with atoms. The pair (a,b) satisfies the following universal property: (a,b) supports both a and b, and if y is another element with this property, then y supports (a,b).

We demonstrate the existence of an element denoted a\land b, with a dual property: a\land b is supported both by a and by b, and if y is another element with this property, then y is supported by a\land b.  The construction does not require any assumptions on the symmetry.

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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.