In this paper, we first proposed the extension principles of neutrosophic multi-sets and cut sets which are a bridge between neutrosophic multi-sets and crisp sets. Then the representation theorem of neutrosophic multi-sets based on cut sets are studied. Finally, the addition, subtraction, multiplication and division operations over neutrosophic multi-sets are defined based on the extension principle.

Let G be a simple connected graph. The generalized polarity Wiener index of G is defined as the number of unordered pairs of vertices of G whose distance is k. Some formulas are obtained for computing the generalized polarity Wiener index of the Cartesian product and the tensor product of graphs in this article.

This article aims to provide a method for defining L-fuzzy algebraic substructures on general algebras. Concretely, the properties of L-fuzzy sets are first reviewed, and their representations are then provided. Next, algebraic substructures are generalised as the closure systems on the power set of the algebra, and the properties of the prime and maximal elements in the above closure system are investigated. Based on these facts, L-fuzzy algebraic substructures with respect to the closure system are defined and studied. Two equivalence characterisations of the sup property of the ordered set L are provided using L-fuzzy substructures. Similarly, some properties of L-fuzzy prime and maximal substructures with respect to the closure system are discussed. Finally, to demonstrate the broad applicability of the theory of L-fuzzy algebraic substructures, the theory is applied to some specific algebraic structures, such as groups and pseudo MV-algebras.

The theory of algebraic frames for a Hilbert space H is a generalization of the theory of frames and generalized frames. The paper applies the theory of unbounded operators to de, ne the dual of algebraic frames with densely de, ned unbounded analysis operators. It is shown that every algebraic frame has an algebraic dual frame, and if an algebraic frame has a nonzero redundancy, then it is not Riesz-type. An example of an algebraic frame with , nite redundancy is constructed which is not a Riesz-type algebraic frame. Finally, for a lower bounded analytic frame, the discreteness of its indexing measure space and the uniqueness of its algebraic dual are studied and shown to be interrelated.

In 2006, Ryu and Takayanagi (RT) pointed out that (with a suitable cutoff) the entanglement entropy between two complementary regions of an equal-time surface of a d+1-dimensional conformal field theory on the conformal boundary of AdS{d+2} is, when the AdS radius is appropriately related to the parameters of the CFT, equal to 1/4G times the area of the $d$-dimensional minimal surface in the AdS bulk which has the junction of those complementary regions as its boundary, where $G$ is the bulk Newton constant. (More precisely, RT showed this for d=1 and adduced evidence that it also holds in many examples in d>1. ) We point out here that the RT-equality implies that, in the quantum theory on the bulk AdS background which is related to the boundary CFT according to Rehren's 1999 algebraic holography theorem, the entanglement entropy between two complementary bulk Rehren wedges is equal to one 1/4G times the (suitably cut off) area of their shared ridge. (This follows because of the geometrical fact that, for complementary ball-shaped regions, the RT minimal surface is precisely the shared ridge of the complementary bulk Rehren wedges which correspond, under Rehren's bulk-wedge to boundary double-cone bijection, to the complementary boundary double-cones whose bases are the RT complementary balls. ) This is consistent with the Bianchi-Meyers conjecture--that, in a theory of quantum gravity, the entanglement entropy, S between the degrees of freedom of a given region with those of its complement takes the form S = A/4G (plus lower order terms)--but only if the phrase `degrees of freedom' is replaced by `matter degrees of freedom'. It also supports related previous arguments of the author--consistent with the author's `matter-gravity entanglement hypothesis'--that the AdS/CFT correspondence is actually only a bijection between just the matter (i. e. non-gravity) sector operators of the bulk and the boundary CFT operators.

Let v be a henselian valuation on a field K, and v ̃,be its unique extension to the algebraic closure K ̃,of K. An element α, ∈, K ̃, K has a distinguished pair if the corresponding set M(α, , K) (defined as the following) has a maximum element M(α, , K)={v ̃, (α,-β, )┤, β,in K ̃, , [K(β, ) ∶, K]<[K(α, ) ∶, K]}. In this case, a pair (α, , β, ) of elements of K ̃,is a distinguished pair for α,whenever β,is an element of smallest degree over K such that deg⁡, α, >deg⁡, β,and v ̃, (α,-β, )=supM(α, , K). In this paper, we first present some results about distinguished pairs of algebraic elements of arbitrary degree over henselian valued fields. Then considering the importance of algebraic elements of prime degree in the extensions of valued fields, we concentrate on such elements. In particular, for α, ∈, K ̃,of prime degree over K, we give a necessary and sufficient condition for the existence of the maximum of the corresponding set M(α, , K) by using the minimal polynomial of α,over K.