In this paper, we introduce the notion of primal topological group, which is a generalization of topological group by a primal. We discuss the characterizations of primal topological groups with illustrative examples.

The main purpose of this paper is to introduce a concept of L- fuzzifying topological groups (here L is a completely distributive lattice) and discuss some of their basic properties and the structures. We prove that its corresponding L-fuzzifying neighborhood structure is translation invariant. A characterization of such topological groups in terms of the corresponding L- fuzzifying neighborhood structure of the unit is given. It is shown that the category of L-fuzzifying topological groups L-FYTPG is topological over the category of groups GRP with respect to the forgetful functor. As an application, the conclusion that the product of L-fuzzifying topological groups is also an L-fuzzifying topological group is proved. Finally, it is proved the forgetful functor preserves the product.

Purpose: The purpose of this paper is to introduce several notions, such as soft topological soft groups, soft topological soft normal subgroups, and soft topological soft factor groups, and to study their properties.Methods: We have adopted the analytical method.Results: We have studied properties of soft topological soft groups, soft subgroups, soft normal subgroups, soft factor groups, and soft homomorphisms.Conclusions: The fundamental homomorphism theorem is extended in a soft topological soft group setting.

A many identities group (MI-group, for short) is an algebraic structure which is generalized a monoid with cancellation laws and is endowed with an invertible anti-automorphism representing inversion. In other words, an MI-group is an algebraic structure generalizing the group concept, except most of the elements have no inverse element. The con-cept of a topological MI-group, as a preliminary study, in the paper " Topological MI-group: Initial study" was introduced by M. Holacapek and N. Skorupova, and we have given a more comprehensive study of this concept in our two recent papers. This article is a continuation of the effort to develop the theory of topological MI-groups and is focused on the study of separation axioms and the isomorphism theorems for topological MI-groups. Moreover, some conditions under which a MI-subgroup is closed will be investigated, and finally, the existence of nonnegative invariant measures on the locally compact MI-groups are introduced.

Let (X,+)(X,+) be a topological abelian group. We discuss regularity of solutions f:X®Rf:X®R of Hlawka’s functional inequality f(x+y)+f(y+z)+f(x+z)£f(x+y+z)+f(x)+f(y)+f(z),f(x+y)+f(y+z)+f(x+z)≤f(x+y+z)+f(x)+f(y)+f(z), postulated for all x,y,z∈Xx,y,z∈X. We study the lower and upper hull of ff. Moreover, we provide conditions which imply continuity of ff. We prove, in particular, that if XX is generated by any neighborhood of zero, f is continuous at zero, and f(0)=0f(0)=0, then f is continuous on X.

In this study, we investigate the further properties of quasi irresolute topological groups defined in [20]. We show that if a group homomorphism ¦ between quasi irresolute topological groups is irresolute at eG, then ¦ is irresolute on G. Later we prove that in a semi-connected quasi irresolute topological group (G; *; Ƭ), if V is any symmetric semi-open neighborhood of eG, then G is generated by V. Moreover it is proven that a subgroup H of a quasi-irresolute topological group (G; *; Ƭ) is semi-discrete if and only if it has a semi-isolated point.

Suppose that G is a compact Hausdorff topological group and H is a closed subgroup of G. The relative commutativity degree of H in G, denoted by Pr(H,G), represents the probability that an element of H commutes with an element of G. Let D(G) be the set of all relative commutativity degrees of subgroups of G. In this paper, we will study the structure of topological groups that have exactly three relative commutativity degrees for their subgroups. In particular, we will show that for such groups, the centralizer of every non-central element is a maximal abelian subgroup. We will also provide examples of groups that have three relative commutativity degrees.