# Question on free Boolean algebras

Every Boolean algebra $A$ is isomorphic to a field of set. In particular, if $A$ is finite, then $A$ is isomorphic to the power set of its atoms.

Now, suppose that $A$ is free Boolean algebra with 2 free generators (or atoms). Because $A$ is a finite Boolean algebra, it is isomorphic to the power set of its atoms. However, $A$ has $16$ elements, whereas the power set of its atoms has only 4 elements.

Could anybody explain what I seem to miss.

• Actually two generators x and y give four atoms (x and y, x and not y, not x and y, not x and not y). – David C. Ullrich Sep 11 '15 at 14:20
• @DavidC.Ullrich I think your comment should be an answer. – Alex Kruckman Sep 14 '15 at 23:49

In particular, if $B$ is freely generated by $\{x,y\}$, it turns out that $B$ has four atoms, namely $x\wedge y$, $\neg x\wedge y$, $x\wedge \neg y$, and $\neg x\wedge\neg y$. More generally, the free Boolean algebra on a finite set with $n$ elements has $2^n$ atoms, namely all the ways to form meets of the generators or their complements.
{"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": 1, "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, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9104358553886414, "perplexity": 221.69424559718885}, "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": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560627998580.10/warc/CC-MAIN-20190617203228-20190617225228-00040.warc.gz"}