Ngcibi, Murali and Makamba [Fuzzy subgroups of rank two abelian$p$-group, Iranian J. of Fuzzy Systems {bf 7} (2010), 149-153]considered the number of fuzzy subgroups of a finite abelian$p$-group $mathbb{Z}_{p^m}times mathbb{Z}_{p^n}$ of rank two, andgave explicit formulas for the cases when $m$ is any positiveinteger and $n=1,2,3$. Even though their method can be used for thecases when $n=4,5,l...

We study HKT structures on nilpotent Lie groups and on associated nilmanifolds. We exhibit three weak HKT structures on R which are homogeneous with respect to extensions of Heisenberg type Lie groups. The corresponding hypercomplex structures are of a special kind, called abelian. We prove that on any 2-step nilpotent Lie group all invariant HKT structures arise from abelian hypercomplex struc...

We study topological von Neumann regularity and principal von Neumann regularity of Banach algebras. Our main objective is comparing these two types of Banach algebras and some other known Banach algebras with one another. In particular, we show that the class of topologically von Neumann regular Banach algebras contains all $C^*$-algebras, group algebras of compact abelian groups and ...

We study algebraic properties of categories of Merotopic, Nearness, and Filter Algebras. We show that the category of filter torsion free abelian groups is an epireflective subcategory of the category of filter abelian groups. The forgetful functor from the category of filter rings to filter monoids is essentially algebraic and the forgetful functor from the category of filter groups to the cat...

If M and Nare ordered groups, we denote by Mx N the group MxNequipped with the lexicographical order: (a,b) < (a',b') if and only if a < a' or {a = a' and b < b'). In [1], A. L. S. Corner gives an example of two countable abelian groups A, G such that G and A x A x G are isomorphic while G and A x G are not isomorphic; he also gives an example of two nonisomorphic countable abelian groups G, H ...

The purpose of this note is to record for my algebra class a proof of the Inverse Galois Problem for finite abelian groups. Recall that the Inverse Galois Problem is stated as follows: Given a finite group G, is there a Galois extension Q ⊆ K such Gal(K/Q) = G? The crucial point in the problem is that the base field is Q, since given any finite group G, there is a Galois extension of fields F ⊆...

In 1937, Baer [5] introduced the notion of the type of an element in a torsion-free abelian group and showed that this notion provided a complete invariant for the classification problem for torsion-free abelian groups of rank 1. Since then, despite the efforts of such mathematicians as Kurosh [23] and Malcev [25], no satisfactory system of complete invariants has been found for the torsion-fre...

In 1937, Baer [5] introduced the notion of the type of an element in a torsion-free abelian group and showed that this notion provided a complete invariant for the classification problem for torsion-free abelian groups of rank 1. Since then, despite the efforts of such mathematicians as Kurosh [23] and Malcev [25], no satisfactory system of complete invariants has been found for the torsion-fre...