Is there a common name for pairs $(X,\alpha)$, where $X$ is a set and $\alpha : X \times X \to X$ is a bijection? Once I have heard "heap" for this, but this already has a different meaning. Notice: These pairs consistute an algebraic category, so that there are colimits. They are not so easy to construct explicitly, though. Any reference is appreciated.

  • $\begingroup$ In [choiceless] set theory these are called idemmultiples. But usually that refers to the cardinal of $X$ and we are not really fussy about the bijection. $\endgroup$ – Asaf Karagila Dec 12 '15 at 10:59
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    $\begingroup$ According to nLab they are called Jónsson-Tarski algebra $\endgroup$ – Hanul Jeon Dec 12 '15 at 11:00
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    $\begingroup$ @HanulJeon: Thank you! Please make this an answer. (After all, it is the answer.) $\endgroup$ – Martin Brandenburg Dec 12 '15 at 11:03

Such objects are called Jónsson-Tarski algebra, according to nLab.

  • $\begingroup$ Thank you! An alternative name is Cantor algebra. There is even a Wikipedia article. en.wikipedia.org/wiki/J%C3%B3nsson%E2%80%93Tarski_algebra $\endgroup$ – Martin Brandenburg Dec 12 '15 at 11:08
  • $\begingroup$ I like the term "Cantor algebra" more, since Cantor invented pairing functions (right?), and Jónsson-Tarski study these algebras only on one page, whereas Smirnov and others have several publications on Cantor algebras. $\endgroup$ – Martin Brandenburg Dec 12 '15 at 12:03
  • $\begingroup$ Can you perhaps explain what the nlab means with the sentence "Loosely speaking, a Jónsson-Tarski algebra is an isomorphism $2^{\aleph_0} \cong 2^{\aleph_0} \times 2^{\aleph_0}$ gone algebra."? Perhaps my English is not good enough. $\endgroup$ – Martin Brandenburg Dec 14 '15 at 10:31
  • $\begingroup$ @MartinBrandenburg My English is also not good enough, so I just can say that it might intend the explanation given below. $\endgroup$ – Hanul Jeon Dec 14 '15 at 10:34

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