In this paper, we begin to study the subalgebra lattice of a Leibniz algebra. particular, deal with algebras whose is modular, upper semi-modular, lower distributive, or dually atomistic. The fact that non-Lie algebra has fewer one-dimensional subalgebras in general results number conditions being weaker than Lie case.

A Poisson geometry arising from maximal commutative subalgebras is studied. A spectral sequence convergent to Hochschild homology with coefficients in a bimodule is presented. It depends on the choice of a maximal commutative subalgebra inducing appropriate filtrations. Its E p,q -groups are computed in terms of canonical homology with values in a Poisson module defined by a given bimodule and ...

We introduce a new notion of a sufficient subalgebra for quantum states: a subalgebra is 2sufficient for a pair of states {ρ0, ρ1} if it contains all Bayes optimal tests of ρ0 against ρ1. In classical statistics, this corresponds to the usual definition of sufficiency. We show this correspondence in the quantum setting for some special cases. Furthermore, we show that sufficiency is equivalent ...

Abstract. In this paper we introduce doubly symmetric functions, arising from the equivalence of particular linear combinations of Schur functions and hook Schur functions. We study algebraic and combinatorial aspects of doubly symmetric functions, in particular as they form a subalgebra of the algebra of symmetric functions. This subalgebra is generated by the odd power sum symmetric functions...

In this paper, we take a GL-quantale as the truth value table to study a new rough set model—L-valued fuzzy rough sets. The three key components of this model are: an L-fuzzy set A as the universal set, an L-valued relation of A and an L-fuzzy set of A (a fuzzy subset of fuzzy sets). Then L-valued fuzzy rough sets are completely characterized via both constructive and axiomatic approaches. 

We inquire further into the properties on fuzzy closed ideals. We give a characterization of a fuzzy closed ideal using its level set, and establish some conditions for a fuzzy set to be a fuzzy closed ideal. We describe the fuzzy closed ideal generated by a fuzzy set, and give a characterization of a finite-valued fuzzy closed ideal. Using a t-norm T , we introduce the notion of (imaginable) T...

Perhaps the most celebrated topological completeness result in modal logic is the McKinseyTarski theorem that if we interpret modal diamond as topological closure, then S4 is complete for the real line or indeed any dense-in-itself separable metrizable space [10]. This result was proved before relational semantics for modal logic was introduced. In the last 15 years, utilizing relational semant...

We prove a spectral theorem for bimodules in the context of graph C∗-algebras. A bimodule over a suitable abelian algebra is determined by its spectrum (i.e., its groupoid partial order) iff it is generated by the Cuntz– Krieger partial isometries which it contains iff it is invariant under the gauge automorphisms. We study 1-cocycles on the Cuntz–Krieger groupoid associated with a graph C∗-alg...