**Jeong-Gon Lee 1,\*, Young Bae Jun <sup>2</sup> and Kul Hur <sup>1</sup>**


Received: 1 August 2020; Accepted: 25 August 2020; Published: 28 August 2020

**Abstract:** In this paper, we define the notions of *i*-octahedron groupoid and *i*-OLI [resp., *i*-ORI and *i*-OI], and study some of their properties and give some examples. Also we deal with some properties for the image and the preimage of *i*-octahedron groupoids [resp., *i*-OLI, *i*-ORI and *i*-OI] under a groupoid homomorphism. Next, we introduce the concepts of *i*-octahedron subgroup and normal subgroup of a group and investigate some of their properties. In particular, we obtain a characterization of an *i*-octahedron subgroup of a group. Finally, we define an *i*-octahedron subring [resp., *i*-OLI, *i*-ORI and *i*-OI] of a ring and find some of their properties. In particular, we obtain two characterizations of *i*-OLI [resp., *i*-ORI and *i*-OI] of a ring and a skew field, respectively.

**Keywords:** octahedron set; *i*-octahedron subgroupoid; *i*-octahedron ideal; *i*-sup-property, *i*-octahedron subgroup; *i*-octahedron subring

**MSC:** 20N25
