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We survey the state of the art on amalgamation in varieties of semilinear residuated lattices. Our discussion emphasizes two prominent cases from which much insight into the general picture may be gleaned: idempotent varieties and their…
This paper studies the concept of stable perturbation $B\in\mathbb{C}^{n\times n}$ for the core-EP inverse of a matrix $A\in\mathbb{C}^{n\times n}$ with index $k$. For a given stable perturbation $B$ of $A$, explicit expressions of its…
A map $f\colon R\to S$ between (associative, unital, but not necessarily commutative) rings is a\emph{brachymorphism} if $f(1+x)=1+f(x)$ and $f(xy)=f(x)f(y)$ whenever $x,y\in R$. We tackle the problem whether every brachymorphism is…
In the first part of the paper, some extensions of the classical Dynkin-Specht-Wever lemma are developed. In the second part, we extend Burgunder's splitting construction, and relate back to the Kashiwara-Vergne problem.
Given a module $X$ and a regular cardinal $\kappa$ we study various notions of $(\kappa,\mathrm{Add}(X))$-freeness and $(\kappa,\mathrm{Add}(X))$-separability. Bearing on appropriate set-theoretic assumptions, we construct a non-trivial…
We introduce a new class of division algebras, the hyperpolyadic algebras, which correspond to the binary division algebras $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$, $\mathbb{O}$ without considering new elements. First, we use the matrix…
Variation diminution (VD) is a fundamental property in total positivity theory, first studied in 1912 by Fekete-P\'olya for one-sided P\'olya frequency sequences, followed by Schoenberg, and by Motzkin who characterized sign regular (SR)…
The objective of this paper is to give alternative proofs for the symmetric Poincar\'e-Birkhoff-Witt theorem utilizing the Magnus recursion formulae or Dynkin's non-commutative polynomial comparison method and simple universal algebraic…
In this article, we introduce an exponential for tropical matrices and show that this series is essential for the analysis of certain kinds of stability in discrete event dynamic systems. A notion of a generalised eigenvector is introduced…
The Algebraic Kirchberg-Phillips Question for Leavitt path algebras asks whether unital $K$-theory is a complete isomorphism invariant for unital, simple, purely infinite Leavitt path algebras over finite graphs. Most work on this problem…
Suppose that $(\mathcal{F},\mathcal{M})$ is an injective structure of $R$-Mod such that the class $\mathcal{F}$ is closed for direct limits, then two modules in $\mathcal{M}$ are isomorphic if there are maps in $\mathcal{F}$ from each one…
This paper provides some new characterizations of the diamond partial order for rectangular matrices by using properties of inner inverses, minus order, and SVD decompositions. In addition, the recently introduced 1MP generalized inverse…
We present techniques for computing Gerstenhaber brackets on Hochschild cohomology of general twisted tensor product algebras. These techniques involve twisted tensor product resolutions and are based on recent results on Gerstenhaber…
We classify the 6-dimensional Lie algebras that can be endowed with an abelian complex structure and parameterize, on each of these algebras, the space of such structures up to holomorphic isomorphism.
In this paper, we establish a connection between evolution algebras of dimension two and Hopf algebras, via the algebraic group of automorphisms of an evolution algebra. Initially, we describe the Hopf algebra associated with the…
A magma is called equidecomposable when the operation is injective, or, in other words, if $x+y=x'+y'$ implies that $x=x'$ and $y=y'$. A magma is free iff it is equidecomposable and graded, hence the notion of equidecomposability is very…
We survey free magmas and we explore the structure of their submagmas. By equipping the cyclic free magma with a second distributive operation we obtain a ringoid-like structure with some primitive arithmetical properties. A submagma is…
Recently, Amnon Neeman settled a bold conjecture by Antieau, Gepner, and Heller regarding the relationship between the regularity of finite-dimensional noetherian schemes and the existence of bounded $t$-structures on their derived…
Let $R$ be a noncommutative ring, and let $S$ be an $m$-system of $R$. In this paper, we give more results on the concept of almost prime (right) ideals, that were introduced by the first two authors, especially in (right) $S$-unital rings,…
In this paper, we first introduce the concept of Rota-Baxter family BiHom-$\Omega$-associative algebras, then we define the cochain complex of BiHom-$\Omega$-associative algebras and verify it via Maurer-Cartan methods. Next, we further…