Characterization of Polygroups by IP-Subsets

Abstract

In this paper, we define the concept of IP-subsets of a polygroup and single polygroups. Indeed, if 〈P, ◦, 1,−1 〉 is a polygroup of order n, then a non-empty subset Q of P is an IP-subset if 〈Q, ∗, e,I 〉 is a polygroup, where for every x, y ∈ Q, x∗ y = (x◦ y)∩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.