Characterization of polygroups by IP-subsets

D. Heidari ; B. Davvaz

Eurasian mathematical journal, Tome 11 (2020) no. 3, pp. 35-41

Résumé

In this paper, we define the concept of IP-subsets of a polygroup and single polygroups. Indeed, if $\langle P,\circ,1,{}^{-1} \rangle$ is a polygroup of order $n$, then a non-empty subset $Q$ of $P$ is an IP-subset if $\langle Q,*,e,{}^I \rangle$ is a polygroup, where for every $x, y\in Q$, $x*y=(x\circ y)\cap Q$. If $P$ has no IP-subset of order $n-1$, then it is single. We show that every non-single polygroup of order $n$ can be constructed from a polygroup of order $n-1$. In particular, we prove that there exist exactly $7$ single polygroups of order less than $5$.