## Return to Question

4 edited tags

3 added 334 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 $G$.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 $G$.

1

# Antichains and measure-preserving actions on Boolean algebras

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)?
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 46, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9619230628013611, "perplexity_flag": "head"}