Kalicki showed in [5] that both of these problems are decidable. Kalicki’s algorithm is obtained by using a fact that only a bounded number of identities has to be verified to decide whether a finite algebra belongs to a finitely generated variety. This approach to the problem turned out to be very inefficient. Bergman and Slutzki presented in [2] an algorithm that solves the universal membersh...

The dynamics of Vlasov kinetic moments is shown to be Lie-Poisson on the dual Lie algebra of symmetric contravariant tensor fields. The corresponding Lie bracket is identified with the symmetric Schouten bracket and the moment Lie algebra is related with a bundle of bosonic Fock spaces, where creation and annihilation operators are used to construct the cold plasma closure. Kinetic moments are ...

We define two such pairs (X, i), (Y, j) to be equivalent if there is an isogeny α : X → Y such that if α̃ : End(X) ' End(Y ) is the induced isomorphism, we have α̃ ◦ i = j. It is easy to check that this is an equivalence relation. We write X ∼ Y if (X, i) is equivalent to (Y, j). Let AF be the collection of equivalence classes. We will classify such equivalence classes. Let D be any R-algebra. A ...

In the present work the properties of Cartan subalgebras and their connection with regular elements in finite dimensional Lie algebras are extended to the case of Leibniz algebras. It is shown that Cartan subalgebras and regular elements of a Leibniz algebra correspond to Cartan subalgebras and regular elements of a Lie algebra by a natural homomorphism. Conjugacy of Cartan subalgebras of Leibn...

1.1. Linear algebra is assumed as a prerequisite to these notes. However, this section serves to review the language of abstract linear algebra (particularly tensor products) for use in the remainder of these notes. We assume the reader familiar with these notions, and therefore do not provide a complete and detailed treatment. In particular, we assume the reader knows what a homomorphism of al...

The theory of Boolean algebras can be fruitfully applied to judgment aggregation: Assuming universality, systematicity and a sufficiently rich agenda, there is a correspondence between (i) non-trivial deductively closed judgment aggregators and (ii) Boolean algebra homomorphisms defined on the power-set algebra of the electorate. Furthermore, there is a correspondence between (i) consistent com...

The descent algebra Σ(W) is a subalgebra of the group algebra QW of a finite Coxeter group W, which supports a homomorphism with nilpotent kernel and commutative image in the character ring of W. Thus Σ(W) is a basic algebra, and as such it has a presentation as a quiver with relations. Here we construct Σ(W) as a quotient of a subalgebra of the path algebra of the Hasse diagram of the Boolean ...

We define and study the category Rep(Q, F1) of representations of a quiver in Vect(F1) the category of vector spaces ”over F1”. Rep(Q, F1) is an F1–linear category possessing kernels, co-kernels, and direct sums. Moreover, Rep(Q, F1) satisfies analogues of the Jordan-Hölder and Krull-Schmidt theorems. We are thus able to define the Hall algebra HQ of Rep(Q, F1), which behaves in some ways like ...

We study the switching aspects of molecular computing from a novel algebraic point of view. Our approach is based on the concept indexed monoidal algebra, which provides an equivalent formalism for compact closed categories being used recently in the literature in connection with quantum computing. The point is to separate syntax, as the algebra G of graphs, from semantics, which is related to ...

We define three semantic models for actor computation starting with a generalization to open systems of Clinger’s event diagram model, and forming two abstractions: interaction diagrams and interaction paths. An algebra is defined on each semantic domain with operations for parallel composition, hiding of internal actors, and renaming. We use these models to provide semantics for descriptions o...