The notion of ordered semihyperrings is a generalization of ordered semirings and a generalization of semihyperrings. In this paper, the Galois connection between ordered semihyperrings are studied in detail and various interesting results are obtained. A construction of an ordered semihyperring via a regular relation is given. Furthermore, we present the Galois connection between homomorphisms and derivations on an ordered semihyperring.

The main purpose of this paper is to study fundamental properties of weak Γ -hyperfilters on ordered Γ -semihypergroups that is a generalization of Γ -hyperfilters. Also, we investigate the relationship between weak Γ -hyperfilters and prime Γ -hyperideals in ordered Γ -semihypergroups. Finally, we introduce weak m , n - Γ -hyperfilters of ordered Γ -semihypergroups and conclude in paper with arising explicit theorems.

Vague graphs (VGs), belonging to the FGs family, have good capabilities when faced with problems that cannot be expressed by FGs. When an element’s membership is not clear, neutrality is a good option that can be well-supported by a VG. The previous definition limitations in irregular-FG have led us to offer new definitions in VGs. So, in this paper, we introduce the concepts of strongly edged irregular vague graphs (SEIVGs), strongly edged totally irregular vague graphs (SETIVGs), and perfectly regular vague graphs (PRVGs) with several examples. A comparative study between SEIVGs and SETIVGs is presented. Finally, an application of vague influence digraph has been presented.