Some new semiring structures

In this paper, we introduce some new semiring structures by considering the prime factors, sum of divisors and the number of relatively prime positive divisors of a pair of non-negative integers and solve a variety of system of linear equations over one of these semirings. We then discuss the properties of these structures and how they compare with classical algebra. The paper is concluded by providing a key-exchange scheme using one of the newly discovered semirings.

A domain wall in twisted M-theory

We study a four-dimensional domain wall in twisted M-theory. The domain wall is engineered by intersecting D6 branes in the type IIA frame. We identify the classical algebra of operators on the domain wall in terms of a higher vertex operator algebra, which describes the holomorphic subsector of a 4d \mathcal{N}=1𝒩=1 supersymmetric field theory, and compute the associated mode algebra. We conjecture that the quantum deformation of the classical algebra is isomorphic to the bulk algebra of operators from which we establish twisted holography of the domain wall.

M-Hazy Vector Spaces over M-Hazy Field

The generalization of binary operation in the classical algebra to fuzzy binary operation is an important development in the field of fuzzy algebra. The paper proposes a new generalization of vector spaces over field, which is called M-hazy vector spaces over M-hazy field. Some fundamental properties of M-hazy field, M-hazy vector spaces, and M-hazy subspaces are studied, and some important results are also proved. Furthermore, the linear transformation of M-hazy vector spaces is studied and their important results are also proved. Finally, it is shown that M-fuzzifying convex spaces are induced by an M-hazy subspace of M-hazy vector space.

Regular Parameter Elements and Regular Local Hyperrings

Inspired by the concept of regular local rings in classical algebra, in this article we initiate the study of the regular parameter elements in a commutative local Noetherian hyperring. These elements provide a deep connection between the dimension of the hyperring and its primary hyperideals. Then, our study focusses on the concept of regular local hyperring R, with maximal hyperideal M, having the property that the dimension of R is equal to the dimension of the vectorial hyperspace MM2 over the hyperfield RM. Finally, using the regular local hyperrings, we determine the dimension of the hyperrings of fractions.

MatCos 3.X

The authors present a sketch of the Matcos 3.X environment. In particular, they present the axiomatic and genetic commands. There are genetic commands for the following: plane and solid geometry, arithmetic and classical algebra, calculus, and numerical methods, with an outline of real functions with two real variables, descriptive statistics, elements of combinatorics, probability, and linear algebra. Each command will be illustrated with an example. There are also comments and suggestions on the pedagogical and didactic use of some commands. The commands will not be displayed in strictly alphabetical order, but a more logical and consequential line has been preferred. At the end of the chapter, the alphabetical list of all the commands available in the programming environment will be shown.

A note on the support of a hypermodule

In this paper, we initiate the study of the notion of support of a hypermodule over a Krasner hyperring, providing several connections with the annihilator of such hypermodules. We concentrate on the similarities/differences between these concepts and the analogous ones from classical algebra (module theory). After defining and characterizing the support of a hypermodule (and in particular of a finitely generated hypermodule), by using the notion of length of a hypermodule, we determine the support of a quotient hypermodule containing only one maximal hyperideal.