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 $G$.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$is the whole automorphism group, however. 2 added 268 characters in body Let$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\$.