In 2024, the authors developed well-known algorithms for the construction and analysis of finite nearrings, implementing it to compute local nearrings of order greater than 31. The new package was named LocalNR package. In this paper, using our method we indicate seventeen \(2\)-generated groups of exponent 8 which are the additive groups of zero-symmetric local nearrings of order 128; and formulate a conjecture on all such nearrings according to which there left only 2 open cases; prove that there are no local nearrings with the Miller-Moreno multiplicative group of order 64 and the additive group to be a \(2\)-generated group of order 128.

We give a complete description of connected non-negative Dynkin type Dyn\(_I=\mathbb{E}_m\) posets and prove that the number of such posets is finite. Moreover, by means of computer assisted analysis, we give a complete Coxeter classification of this class and prove that the pair (Dyn\(_I=\mathbb{E}_m,\) specc \(_I\)), where specc \(_I\subseteq\mathbb{C}\) denotes the Coxeter spectrum of \(I\), determines \(I\) uniquely, up to the strong Gram \(\mathbb{Z}\)-congruence.

This paper aims to shed light on new (sub)classes of matroids originating from cluster algebras and investigate their properties. We focus on what we call cluster matroids and build some results on them. Then, we point out a relationship between these kinds of matroids and uniform matroids and study their minors.

In this paper we study a connection between variants of semigroups and Brandt semigroups. We find necessary conditions under which a variant of a semigroup is a Brandt semigroup. For variants of Rees matrix semigroups we studied a structure of a sandwich matrix. We proved that if semigroup does not contain a bicyclic subsemigroup, then any variant of this semigroup is not a Brandt semigroup. Thus a variant of a finite semigroup is not a Brandt semigroup.

It is known that if the unit group of an integral group ring ZG is trivial, then the unit group of Z(G × C 2 ) is trivial as well [3] . The aim of this study is twofold: firstly, to identify rings R that are D-adapted for the direct product D = G × H of abelian groups G and H, such that the unit group of the ring R(G × H) is trivial. Our second objective is to investigate the necessary and sufficient conditions on both the ring R and the direct factors of D to satisfy the property that the normalized unit group V (RD) is trivial in the case where D is one of the groups G × C 3 , G × K 4 or G × C 4 , where G is an arbitrary finite abelian group, C n denotes a cyclic group of order n and K 4 is Klein 4-group. Hence, the study extends the related result in [18] . The author wishes to express gratitude to the referees for their valuable suggestions and to all the members of the Algebra and Discrete Mathematics community for their support and contributions.