We study unit groups of weighted Leavitt path algebras. Let $K$ be a field of characteristic $0$ and let $(E,ω)$ be a finite connected weighted graph. We prove that $L_K(E,ω)^\times$ is abelian if and only if $L_K(E,ω)$ is a domain. Equivalently, $L_K(E,ω)^\times$ contains no non-cyclic free subgroup if and only if $L_K(E,ω)$ is a domain.

Let $n$ be a positive integer, $n \not=1$, and $S_n$ the set of all $n \times n$ real symmetric matrices. A nonempty subset $\U \subset S_n$ is called a matrix domain if it is open and connected and a map $ϕ: \U \to S_n$ is said to be an order emebedding if for every pair $X,Y \in \U$ we have $X \le Y \iff ϕ(X) \le ϕ(Y)$. We describe the general form of such maps.

In this note, we prove that a square matrix of size $n$ over a field containing at least $2n$ elements can be expressed as the product of two matrices similar to companion matrices, that is to say matrices with the same minimal and characteristic polynomial, if and only if the rank of $A$ is greater than $n-2$, using only elementary facts. We will also give some partial results valid over smaller fields.

Let $L=K(θ)\simeq K[x]/f(x)$ be a simple field extension in prime characteristic $p>0$, $L^{sep}$ and $L^{pi}$ be the maximal separable and purely inseparable subfields of $L$, respectively. Let $N/K$ be a purely inseparable field extension. For the field extensions $L/K$ and $NL/N$, the aim of the paper is to give explicit descriptions of the following subfields and their degrees in terms of the coefficients of the polynomial $f$ and two numerical field invariants $m_f$ and $m_{f,N}$: $L^{pi}$, $L^{pi}L^{sep}$, $(NL)^{pi}$ and $(NL)^{pi}(NL)^{sep}$. From these results, we derive new explicit criteria for $L=L^{pi}L^{sep}$ and $NL=(NL)^{pi}(NL)^{sep}$.

In this paper, we first introduce Nijenhuis Lie 2-algebras as the categorification of Nijenhuis Lie algebras. We prove that the category of Nijenhuis Lie 2-algebras is equivalent to the category of 2-term Nijenhuis $L_\infty$-algebras. Next, given a Nijenhuis Lie algebra, we introduce the notion of a 2-representation and show that the corresponding semidirect product inherits a Nijenhuis Lie 2-algebra structure. On the other hand, we consider a $2$-term representation up to homotopy of a Nijenhuis Lie algebra and obtain a $2$-term Nijenhuis $L_\infty$-algebra as the semidirect product. Finally, we show that the category of $2$-representations and the category of $2$-term representations up to homotopy of a Nijenhuis Lie algebra are equivalent.

Recently, Ehret and Gilliers introduced the notion of a (trivially) restricted post-Lie algebra, recovering the concepts of a restricted Lie algebra and a restricted pre-Lie algebra. In this paper, we specifically introduce restricted Rota-Baxter Lie algebras of arbitrary weight with an intrinsic graph subalgebra characterization. We show that, via the splitting property, they give rise to restricted post-Lie algebras, and furthermore possess a novel replication property. We then present two natural constructions of such restricted Rota-Baxter structures in prime characteristic: one arising from Rota-Baxter associative algebras of arbitrary weight, and the other from Rota-Baxter Lie algebras of weight $1$. The Rota-Baxter $p$-envelopes of a Rota-Baxter Lie algebra are also examined.

We give a new and elementary construction of primitive positive decomposition of higher arity relations into binary relations on finite domains. Such decompositions come up in applications to constraint satisfaction problems, clone theory and relational databases. The construction exploits functional completeness of 2-input functions in many-valued logic by interpreting relations as graphs of partially defined multivalued 'functions'. The 'functions' are then composed from ordinary functions in the usual sense. The construction is computationally effective and relies on well-developed methods of functional decomposition, but reduces relations only to ternary relations. An additional construction then decomposes ternary into binary relations, also effectively, by converting certain disjunctions into existential quantifications. The result gives a uniform proof of Peirce's reduction thesis on finite domains, and shows that the graph of any Sheffer function composes all relations there.

Any linear space of square matrices has an associated eigenvector variety. Its points are eigenvectors of matrices from that linear space. We present a systematic study of eigenvector varieties, with focus on Lie algebras and Hamiltonians of quantum systems.

We study Rota--Baxter operators on vertex algebras using the integrated $λ$-bracket formalism. A Rota--Baxter operator produces a deformed vertex algebra structure, and the difference between the deformed and original brackets yields a two-cocycle in vertex algebra cohomology. This generalizes the classical relation between Rota--Baxter operators and Hochschild two-cocycles. We also characterize when this two-cocycle is trivial, showing that non-scalar operators give rise to non-trivial cohomology classes.

Let $\mathbb{F}$ be an algebraically closed field of characteristic $0$. Given a square matrix $A \in \mathbb{F}^{n \times n}$ and a polynomial $f \in \mathbb{F}[w]$, we determine the Jordan canonical form of the formal Fréchet derivative of $f(A)$, in terms of that of $A$ and of $f$. When $\mathbb{F}\subseteq \mathbb{C}$, via Hermite interpolation, our result provides a solution to [N.J. Higham, \emph{Functions of Matrices: Theory and Computation}, Research Problem 3.11]. A generalization consists of finding the Jordan canonical form of linear combinations of Kronecker products of powers of two square matrices, i.e., $\sum_{i,j} a_{ij} (X^i \otimes Y^j)$. For this generalization, we provide some new partial results, including a partial solution under certain assumptions and general bounds on the number and the sizes of Jordan blocks.

We show that the monoids totM_{k,1} introduced by Birget and their generalizations tot nM_{k,r} which extend the Brin-Higman-Thompson groups, can be realized as the endomorphism monoids of higher-dimensional Jónsson-Tarski algebras. We also show how elements of these monoids can be thought of as "rewrite rules". We use these representations to show that the monoids are finitely presented.

Over an arbitrary field, we conduct a comprehensive study of the polynomial identities and codimensions of two- and three-dimensional metabelian non-Lie Leibniz algebras. In addition, we compute the images of multihomogeneous polynomials on two-dimensional Leibniz algebras and, as a consequence, prove that the image of any multilinear polynomial evaluated on such algebras is always a vector space. Our analysis includes the three nontrivial isomorphism classes in dimension two and the ten isomorphism classes in dimension three, all of which are metabelian. In particular, we determine finite bases for their corresponding $T$-ideals and provide explicit bases for the associated relatively free graded algebras.

We provide a structural analysis for McCarthy algebras, the variety generated by the three-element algebra defining the logic of McCarthy (the non-commutative version of Kleene three-valued logics). Our analysis will be conducted in a very general algebraic setting by introducing McCarthy algebras as a subvariety of unital bands (idempotent monoids) equipped with an involutive (unary) operation $'$ satisfying $x''\approx x$; herein referred to as i-ubands. Prominent (commutative) subvarieties of i-ubands include Boolean algebras, ortholattices, Kleene algebras, and involutive bisemilattices, hence i-ubands provides an algebraic common ground for several non-classical logics. Our main contributions consist in providing for McCarthy algebras: reduced and equivalent axiomatizations; a semilattice decomposition theorem; and representations as certain decorated posets from which the algebraic structure can be uniquely determined.

In this paper, we study how to find rational motions that move a line along a given rational ruled surface. Our goal is to find motions with the lowest possible degree using dual quaternions. While similar problems for point trajectories are well known, the case of line trajectories is more complicated and has not been studied. We explain when such motions exist and how to compute them. Our method gives explicit formulas for constructing these motions and shows that, in many cases, the solution is unique. We also show examples and explain how to use these results to design simple mechanisms that move a line in the desired way. This work helps to better understand the relationship between rational motions and ruled surfaces and may be useful for future research in mechanism design.

Andréka and Maddux classified the relation algebras with at most 3 atoms, and in particular they showed that all of them are representable. Hirsch and Cristiani showed that the network satisfaction problem (NSP) for each of these algebras is in P or NP-hard. The literature contains many results on representations of relation algebras; in particular, some relation algebras with four atoms are not representable. We extend the result of Cristiani and Hirsch to relation algebras with at most 4 atoms: the NSP is always either in P or NP-hard. To this end, we construct universal, fully universal, or even normal representations for these algebras, whenever possible.

In this paper, we give an axiomatic characterization of the cap product in the Hochschild theory of associative unital algebras which are projective over a commutative unital ring. We also give an interpretation of the cap product with coefficients in the algebra via chain maps. We illustrate these results by computing the cap product for truncated polynomial algebras $k[x]/(x^N)$ and for polynomial algebras, where it is identified with the contraction of differential forms by polyvector fields.