In this paper, in addition to some elementary facts about the ultra-groups, which their structure based on the properties of the transversal of a subgroup of a group, we focus on the relation between a group and an ultra-group. It is verified that every group is an ultra-group, but the converse is not true generally. We present the conditions under which, for every normal subultra-group of an u...

In this paper, we first introduce the concepts of G-systems, quotient G-systems and isomorphism theorems on G-systems of n-ary semihypergroups . Also we consider the Green's equivalences on G-systems and further in-vestigate some of their properties. A number of n-ary semihypergroups are constructed and presented as examples in this paper.

We consider presentations that were derived in [3] for the interval groups associated with proper quasi-Coxeter elements of Coxeter group W(Dn). use combinatorial methods to derive alternative groups, and these new show a element W(Dn) cannot be isomorphic Artin type Dn. While specific problems we solve arise from study their solution provides an illustration how techniques indicated by computa...

The concepts of $L$-convex systems and Scott-hull spaces are proposed on frame-valued setting. Also, we establish the categorical isomorphism between  $L$-convex systems and Scott-hull spaces. Moreover, it is proved that the category of $L$-convex structures is  bireflective in the category of $L$-convex systems. Furthermore, the quotient systems of $L$-convex systems are studied.