Given a connected linear algebraic group G over an algebraically closed field of characteristic zero, we describe the subgroup of G generated by all semisimple elements.

The existence of perfect state transfer (PST) on quantum spin networks is a fundamental problem in mathematics and physics. Various works in the literature have explored PST in graphs with arithmetic origins, such as gcd-graphs over [Formula: see text] and cubelike graphs. In this article, building on our recent work on gcd-graphs over an arbitrary finite Frobenius ring, we investigate the existence of PST on these graphs. Our approach is algebraic in nature, enabling us to unify various existing results in the literature.

We determine the irreducible representations of the simple Lie algebras with maximum weight multiplicity 2.

A Pisot numeration system [Formula: see text] for [Formula: see text] is one where the sequence of positive integers [Formula: see text], with [Formula: see text], is generated by a recurrence whose polynomial is the minimal polynomial of a Pisot number. The Zeckendorf numeration [Formula: see text] is the simplest such example. We define generalised equations of [Formula: see text]-Mahler type, and we show that if a sequence over a commutative ring is [Formula: see text]-regular, then it is the sequence of coefficients of a series which is a solution of a [Formula: see text]-Mahler equation. Conversely, if the [Formula: see text]-Mahler equation is isolating, then its solutions define [Formula: see text]-regular sequences. This is a generalisation of results of Becker and Dumas. We provide an example to show that there exist non-isolating [Formula: see text]-Mahler equations whose solutions do not define [Formula: see text]-regular sequences. Our proof yields a new construction of weighted automata that generate classical [Formula: see text]-regular sequences. Our results can be generalised to numeration systems generated by recurrences whose characteristic polynomial is the minimal polynomial of a Pisot number.

In this article, we study the existence of unimodular elements and cancellation of projective modules over Rees-like algebras and its various extensions.

In this paper, we investigate the computational complexity of the knapsack problem and subset sum problem for the following tropical algebraic structures. We consider the semigroup of square matrices of size [Formula: see text] with non-negative entries over the max-plus algebra and the semigroup of square matrices of size [Formula: see text] with positive entries over the max-times algebra. We prove that the knapsack problem and the subset sum problem for these structures are [Formula: see text]-complete. We demonstrate that there are pseudo-polynomial algorithms to solve these problems. Also, we show that for the latter semigroup, there are polynomial generic algorithms to solve the knapsack problem and the subset sum problem.

A non-zero unital ring [Formula: see text] is said to be centrally essential if for every nonzero element [Formula: see text] of [Formula: see text], there exist non-zero central elements [Formula: see text] and [Formula: see text] with [Formula: see text]. In the paper, almost fully prime centrally essential rings are described in terms of ideal extensions, centrally essential Dorroh extensions and trivial extensions.

Recently, in weighted automata theory the weight structure of strong bimonoids has found much interest; they form a generalization of semirings and are closely related to near-semirings studied in algebra. Here, we define polynomials over a set [Formula: see text] of indeterminates as well as an addition and a multiplication. We show that with these operations, they form a right-distributive strong bimonoid, that this polynomial strong bimonoid is free over [Formula: see text] in the class of all right-distributive strong bimonoids and that it is both left- and right-cancellative. We show by purely algebraic reasoning that two arbitrary terms are equivalent modulo the laws of right-distributive strong bimonoids if and only if their representing polynomials are equivalent by the laws of only associativity and commutativity of addition and associativity of multiplication. We give effective procedures for constructing the representing polynomials. As a consequence, we obtain that the equivalence of arbitrary terms modulo the laws of right-distributive strong bimonoids can be decided in exponential time. Using term-rewriting methods, we show that each term can be reduced to a unique polynomial as normal form. We also derive corresponding results for the free idempotent right-distributive polynomial strong bimonoid over [Formula: see text]. We construct an idempotent strong bimonoid which is weakly locally finite but not locally finite and show an application of it in weighted automata theory.

In this work we study the self-similarity of infinite products of the group of the integers. In [A. C. Dantas, T. M. G. Santos and S. N. Sidki, Self-similar abelian groups and their centralizers, Groups Geom. Dyn. 17 (2023) 1–23], the authors ask if there is a self-similar free abelian group of uncountable rank. On the other hand, in [L. Bartholdi and S. N. Sidki, Self-similar products of groups, Groups Geom. Dyn. 14(1) (2020) 107–115] is asked if the Baer–Specker group [Formula: see text] is self-similar (by a result of R. Baer, this group is not free abelian). We answer both questions positively.

In this paper we study Zappatic surfaces. We give a combinatorial description of whether the Galois cover of degeneration of a surface to a union of planes is simply connected. Additionally, we compute Chern numbers and the signatures of the Galois covers of the surfaces and present applications to Hirzebruch and Veronese surfaces.