Mathematics

Given an approximation space ?�U,θ??, assume that E is the indexing set for the equivalence classes of θ and let R θ denote the collection of rough sets of the form ??X ????, X ¯ ¯ ¯ ¯ ??as a regular double Stone algebra and what I. Dunstch referred to as a Katrinak algebra.[7],[8] We give an alternate proof from the one given in [1] of the fact that if | θ u |>1 ?� u?�U then R θ is a core regular double Stone algebra. Further let C 3 denote the 3 element chain as a core regular double Stone algebra and T P U denote the collection of ternary partitions over the set U . In our Main Theorem we show R θ with | θ u |>1 ?� u?�U to be isomorphic to T P E and C E 3 , with E is an indexing set for θ , and that the three CRDSA's are complete and atomic. We feel this could be very useful when dealing with a specific R θ in an application. In our Main Corollary we show explicitly how we can embed such R θ in T P U , C U 3 , respectively, ???α r : R θ ?�T P U ??C U 3 , and hence identify it with its specific images. Following in the footsteps of Theorem 3. and Corollary 2.4 of [7], we show C J 3 ??R θ for ?�U,θ??the approximation space given by U=J?{0,1} , θ={(j0),(j1)}:j?�J} and every CRDSA is isomorphic to a subalgebra of a principal rough set algebra, R θ , for some approximation space ?�U,θ??. Finally, we demonstrate this and our Main Theorem by expanding an example from [1]. Further, we know a little more about the subalgebras of T P U and C U 3 in general as they must exist for every E that is an indexing set for the equivalence classes of any equivalence relation θ on U satisfying | θ u |>1 ?� u?�U .

We consider when the symmetric algebra of an infinite-dimensional Lie algebra, equipped with the natural Poisson bracket, satisfies the ascending chain condition (ACC) on Poisson ideals. We define a combinatorial condition on a graded Lie algebra which we call Dicksonian because it is related to Dickson's lemma on finite subsets of N k . Our main result is: Theorem. If g is a Dicksonian graded Lie algebra over a field of characteristic zero, then the symmetric algebra S(g) satisfies the ACC on radical Poisson ideals. As an application, we establish this ACC for the symmetric algebra of any graded simple Lie algebra of polynomial growth, and for the symmetric algebra of the Virasoro algebra. We also derive some consequences connected to the Poisson primitive spectrum of finitely Poisson-generated algebras.

We give an overview of a number of Schreier-type extensions of monoids and discuss the relation between them. We begin by discussing the characterisations of split extensions of groups, extensions of groups with abelian kernel and finally non-abelian group extensions. We see how these characterisations may be immediately lifted to Schreier split extensions, special Schreier extensions and Schreier extensions respectively. Finally, we look at weakenings of these Schreier extensions and provide a unified account of their characterisation in terms of relaxed actions.

A matrix always has a full rank submatrix such that the rank of this matrix is equal to the rank of that submatrix. This property is one of the corner stones of the matrix rank theory. We call this property the max-full-rank-submatrix property. Tensor ranks play a crucial role in low rank tensor approximation, tensor completion and tensor recovery. However, their theory is still not matured yet. Can we set an axiom system for tensor ranks? Can we extend the max-full-rank-submatrix property to tensors? We explore these in this paper. We first propose some axioms for tensor rank functions. Then we introduce proper tensor rank functions. The CP rank is a tensor rank function, but is not proper. There are two proper tensor rank functions, the max-Tucker rank and the submax-Tucker rank, which are associated with the Tucker decomposition. We define a partial order among tensor rank functions and show that there exists a unique smallest tensor rank function. We introduce the full rank tensor concept, and define the max-full-rank-subtensor property. We show the max-Tucker tensor rank function and the smallest tensor rank function have this property. We define the closure for an arbitrary proper tensor rank function, and show that it is still a proper tensor rank function and has the max-full-rank-subtensor property. An application of the submax-Tucker rank is also presented.

We prove a categorical duality between a class of abstract algebras of partial functions and a class of (small) topological categories. The algebras are the isomorphs of collections of partial functions closed under the operations of composition, antidomain, range, and preferential union (or 'override'). The topological categories are those whose space of objects is a Stone space, source map is a local homeomorphism, target map is open, and all of whose arrows are epimorphisms.

Given a symmetric Leibniz algebra (L,.) , the product is Lie-admissible and defines a Lie algebra bracket [,] on L . Let G be the connected and simply-connected Lie group associated to (L,[,]) . We endow G with a Lie rack structure such that the right Leibniz algebra induced on T e G is exactly (L,.) . The obtained Lie rack is said to be associated to the symmetric Leibniz algebra (L,.) . We classify symmetric Leibniz algebras in dimension 3 and 4 and we determine all the associated Lie racks. Some of such Lie racks give rise to non-trivial topological quandles. We study some algebraic properties of these quandles and we give a necessary and sufficient condition for {them} to be quasi-trivial.

We give a construction that takes a simple linear algebraic group G over a field and produces a commutative, unital, and simple non-associative algebra A over that field. Two attractions of this construction are that (1) when G has type E 8 , the algebra A is obtained by adjoining a unit to the 3875-dimensional representation and (2) it is effective, in that the product operation on A can be implemented on a computer. A description of the algebra in the E 8 case has been requested for some time, and interest has been increased by the recent proof that E 8 is the full automorphism group of that algebra. The algebras obtained by our construction have an unusual Peirce spectrum.

In this article, we prove a class of normal dilation matrices affirming the Marcus-de Oliveira conjecture.

We exhibit an explicit formula for the cardinality of solutions to a class of quadratic matrix equations over finite fields. We prove that the orbits of these solutions under the natural conjugation action of the general linear groups can be separated by classical conjugation invariants defined by characteristic polynomials. We also find a generating set for the vanishing ideal of these orbits.

We derive a closed-form expression for the kth power of semicirculant matrices by using the determinant of certain matrices. As an application, a closed-form expression for the kth power of r-circulant matrices is also povided.