# Name for sets $X$ equipped with a bijection $X \times X \cong X$

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.

• 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. – Asaf Karagila Dec 12 '15 at 10:59
• According to nLab they are called Jónsson-Tarski algebra – Hanul Jeon Dec 12 '15 at 11:00
• @HanulJeon: Thank you! Please make this an answer. (After all, it is the answer.) – Martin Brandenburg Dec 12 '15 at 11:03

• 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. – Martin Brandenburg Dec 14 '15 at 10:31