Related papers: Conditionally atomless extensions of sigma algebra…
We exhibit an adjunction between a category of abstract algebras of partial functions and a category of set quotients. The algebras are those atomic algebras representable as a collection of partial functions closed under relative…
We characterise algebras commutative with respect to a Yang-Baxter operator (quasi-commutative algebras) in terms of certain cosimplicial complexes. In some cases this characterisation allows the classification of all possible…
We define nonassociative cyclic extensions of degree m of both fields and central simple algebras over fields. If a suitable field contains a primitive mth (resp., qth) root of unity, we show that suitable nonassociative generalized cyclic…
We give examples of pairs of isotopic algebras with non-isomorphic congruence lattices. This answers the question of whether all isotopic algebras have isomorphic congruence lattices.
We extend results related to maximal subalgebras and ideals from Lie to Leibniz algebras. In particular, we classify minimal non-elementary Leibniz algebras and Leibniz algebras with a unique maximal ideal. In both cases, there are types of…
Given an algebra $\mathbf{A}$, and terms $s(x_{1},x_{2},\dots x_{k})$ and $t(x_{1},x_{2},\dots x_{k})$ of the language of ${\mathbf A}$, we say that $s$ and $t$ are {\em separated} in ${\mathbf A}$ iff for all $a_{1},a_{2}\dots a_{k}\in A$,…
A classification of the semisimple subalgebras of the Lie algebra of traceless $3\times 3$ matrices with complex entries, denoted $A_2$, is well-known. We classify its nonsemisimple subalgebras, thus completing the classification of the…
Basic definitions and properties of nearly associative algebras are described. Nearly associative algebras are proved to be Lie-admissible algebras. Two-dimensional nearly associative algebras are classified, and its main classes are…
In this paper we prove that small cancellation rings under some natural restrictions are non-amenable and contain non-commutative free associative algebra.
This work adapts the equivalent definitions of division algebras over a field into multiple types of division algebras in a monoidal category. Examples and consequences of these definitions are then established in various monoidal settings.
In this paper we begin to study the subalgebra lattice of a Leibniz algebra. In particular, we deal with Leibniz algebras whose subalgebra lattice is modular, upper semi-modular, lower semi-modular, distributive, or dually atomistic. The…
We classify all two-dimensional simple algebras (which may be non-associative) over an algebraically closed field. For each two-dimensional algebra $\mathcal{A}$, we describe a minimal (with respect to inclusion) generating set for the…
We investigate from an algebraic and topological point of view the minimal prime spectrum of a universal algebra, considering the prime congruences w.r.t. the term condition commutator. Then we use the topological structure of the minimal…
We develop dualities for complete perfect distributive quasi relation algebras and complete perfect distributive involutive FL-algebras. The duals are partially ordered frames with additional structure. These frames are analogous to the…
We establish results about the second cohomology with coefficients in the trivial module, symmetric invariant bilinear forms and derivations of a Lie algebra extended over a commutative associative algebra without unit. These results…
We give characterizations of unital uniform topological algebras and saturated locally multiplicatively convex algebras by means of multiplicative linear functionals. Some automatic continuity theorems in advertibly complete uniform…
We study graded connected algebras over a field of characteristic zero and give an explicit formula for the cyclic homology of a tensor algebra. By means of a slightly new definition of David Anick's notion "strongly free" we are able to…
A new class of infinite dimensional simple Lie algebras over a field with characteristic 0 are constructed. These are examples of non-graded Lie algebras. The isomorphism classes of these Lie algebras are determined. The structure space of…
Finite versions of W-algebras are introduced by considering (symplectic) reductions of finite dimensional simple Lie algebras. In particular a finite analogue of $W^{(2)}_3$ is introduced and studied in detail. Its unitary and non-unitary,…
We develop a structure theory for nilpotent symplectic alternating algebras. We then give a classification of all nilpotent symplectic alternating algebras of dimension up to 10 over any field. The study reveals a new subclasses of powerful…