Fuzzy sub-hoops based on fuzzy points

Using the belongs to relation (e) and quasi-coincident with relation (q) between fuzzy points and fuzzy sets, the notions of an (e, e)-fuzzy sub-hoop, an (e, ev q)-fuzzy sub-hoop and a (q, ev q) -fuzzy sub-hoop are introduced, and several properties are investigated. Characterizations of an (e, e)-fuzzy sub-hoop and an (e, ev q)-fuzzy sub-hoop are displayed. Relations between an (e, e)-fuzzy sub-hoop, an (e, ev q)-fuzzy sub-hoop and a (q, ev q)-fuzzy sub-hoop are discussed. Conditions for a fuzzy set to be a (q, e vq)-fuzzy sub-hoop are considered, and condition for an (e, ev q)-fuzzy sub-hoop to be a (q, ev q)-fuzzy sub-hoop are provided.