Let $G$ be a group of automorphisms of the countable atomless Boolean algebra $B$. Suppose that every orbit of $G$ on $B$ is an antichain. Does it follow that $G$ preserves a non-zero (probability) measure on $B$?


Does the answer change if we extend $B$ to some complete or $\sigma$-complete algebra, and the action of $G$ extends to one in which orbits are still antichains?


I'm also interested in group actions that satisfy a very different condition: $G$ is a group such that for some (all) $a \in B \setminus \{0,1\}$ and for all $b \in B \setminus \{0,1\}$ there is some $g \in G$ such that $ga < b$. Do such actions have a name and has anything been proved about them (or about groups that have such actions)?


Edit: To clarify, by 'antichain' I just mean a set of pairwise incomparable elements. I didn't know about the stronger meaning used by set theorists. For what it's worth I am mainly interested in using actions to understand algebraic properties of the group $G$, so I probably don't need to consider any exotic algebras of the kind set theorists would find interesting; the most obvious examples, such as the countable atomless Boolean algebra or the standard Borel $\sigma$-algebra, are probably good enough. I definitely do not want to assume that $G$ is the whole automorphism group, however.



Antichains and measure-preserving actions on Boolean algebras
Let $G$ be a torsion group. Then every action of $G$ on every Boolean algebra has only antichains as orbits (indeed $gA<A$ implies $g^nA<A$ for every $n\ge 1$ and this contradicts $g$ being torsion).


Let now $G$ be a non-amenable torsion group (there are several such groups, such as Golod-Shafarevich groups, some of which having Kazhdan's Property T by work of Ershov and Jaikin-Zapirain, or many Burnside groups by work of Adian).


Then the left translation action of $G$ on itself induces an action on the Boolean algebra $2^G$ of subsets, with no invariant probability measure (this is a restatement of non-amenability - I assume that you mean probabilities in the usual sense used in the context of Boolean algebras: just finitely additive).


This example of Boolean algebra is not atomless, but modding out by finite subsets yields an atomless Boolean algebra $2^G/\mathrm{fin}$. A probability measure on $2^G/\mathrm{fin}$ is the same as a probability measure on $2^G$ mapping finite subsets to zero, so does not exist for $G$ non-amenable. (This action can be extended to a complete Boolean algebra, with the same properties.)


