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Algebraic Hyperstructures is a generalization of the concept of algebraic structure. One of the concepts on algebraic hyperstructures is hypergroups. The concept of hypergroups can be generalized into hypergroupoids, hypersemigroups and -groups. Hypergroups and their generalization have applications in science fields such as chemistry, biology, and physics. The focus of discussion in this paper is the application of hypergroups and their generalization in the field of chemistry. In basic chemistry, we know chemical reactions called reduction and oxidation (redox) reactions. The purpose of this paper is to analyze the algebraic hyperstructures contained in redox reactions.

Algebraic hyperstructure is a generalized form of the ordinary algebraic structures. Research on algebraic hyperstructures can focus on the theory and its applications. In this chapter, the discussion will focus on the application of algebraic hyperstructures in chemistry. Algebraic hyperstructures play a role in investigating the types of structures found in a chemical reaction. Furthermore, the concept of algebraic hyperstructures is associated with the concept of NeutroAlgebra so that a new type of structure called NeutroHyperstructures is obtained. In this chapter the types of NeutroHyperstructures contained in some chemical reactions will be investigated along with the application of these NeutroHyperstructures.

Clustering merupakan metode pengelompokan dalam suatu database informasi berdasarkan kondisi tertentu. Penelitian ini menerapkan masalah perhitungan k-means clustering dengan pendekatan perhitungan jarak euclidean dengan perhitungan jarak manhattan. Metode yang terbentuk bertujuan untuk membandingkan dalam segi proses pengerjaan antara perhitungan jarak euclidean dengan perhitungan jarak manhattan. Hasil nya berupa perbandingan perhitungan jarak antara perhitungan jarak Euclidean dengan perhitungan jarak Manhattan dalam segi proses pengerjaan untuk dapat menentukan titik-titik pusat penyebaran penyakit covid dari perbandingan perhitungan jarak Euclidean dan perhitungan jarak Manhattan. Hasil perhitungan yang diperoleh yaitu Perhitungan K-Means dengan pendekatan perhitungan jarak Euclidean diperoleh banyaknya percobaan (iterasi) sejumlah 15 kali,sedangkan dengan menggunakan perhitungan jarak Manhattan diperoleh banyaknya percobaaan (iterasi) sejumlah 7 kali. Maka disimpulkan bahwa dalam segi proses pengerjaan Manhattan lebih cepat bila dibandingan dengan Euclidean. Hasil perhitungan yang didapatkan merupakan hasil perhitungan dari Data Covid-19 di Kota Bekasi sampai dengan tanggal 1 September 2021.

Mahatma and Muchtadi-Alamsyah (2017) introduced the notion of the so-called 𝒰-extension modules which is the generalization of the concept of the extension modules. As the classic extension modules induce a long exact sequence of homologies from a given short exact sequence, there is an analogous result in the 𝒰 version in the form of an exact sequence consists of 𝒰-extension modules. It is just that such sequence has only limited number of nonzero modules while the classic one can contain infinitely many. Mahatma and Hadi (2020) proposed sufficient conditions for the sequence so it can be extended. However, it requires some tight assumptions to be hold. In this paper, we shall give another method of extending the sequence so we may avoid the dependency on the additional assumptions.

Given commutative algebra R and R -module U, Hadi et al constructed the category of supermodules CU whose objects are all pairs of R -modules (N, M) where N is direct sum of U and M contains U. If U and W are R-modules then there were relations between the categories CU and CW under some conditions. For instance, if W is direct sum of U then there exists a covariant functor from CU to CW. Further, if U and W are isomorphic then such functor is a category equivalence. In this paper we investigate the relation between two categories of supermodules over different algebras.

Inspired by the notions of the U-exact sequence introduced by Davvaz and Parnian-Garamaleky in 1999, and of the chain U-complex introduced by Davvaz and Shabani-Solt in 2002, Mahatma and Muchtadi-Alamsyah in 2017 developed the concept of the U-projective resolution and the U-extension module, which are the generalizations of the concept of the projective resolution and the concept of extension module, respectively. It is already known that every element of a first extension module can be identified as a short exact sequence. To the simple, there is a relation between the first extension module and the short exact sequence. It is proper to expect the relation to be provided in the U-version. In this paper, we aim to construct a one-one correspondence between the first U-extension module and the set consisting of equivalence classes of short U-exact sequence.Keywords: Chain U-complex, U-projective resolution, U-extension module

The properties of Max Plus Algebra can be applied to the matrix. Furthermore, the generalized inverses in a matrix can also be related to this algebra. This paper discusses some properties of these concepts on 2 x 2 matrices integers modulo 3. Here, we derive the properties of matrices in this algebra. Then, the generalized inverses in these matrices is formed. We conclude that there is some relation between the matrices and its generalized inverses. Some examples of matrices are given to illustrate the properties.

Let R be commutative algebra with unit element. Given any R-module U, there exists a category CU whose objects are all pairs (N,M) where N is direct sum of U and M contains U. It has been shown that if W is a direct summand of U then there exists a functor from CU to CW. Further, if W and U are isomorphic then there exists category equivalence between CU and C W .

Let R be commutative algebra with unit element. Given any R -modules M and N, and any nonzero submodule U of M, Mahatma and Muchtadi-Alamsyah defined the k -th U -extension module of N by M for every positive integer k. Further, Mahatma showed that given any short exact sequence of R -modules and R -module homomorphisms, there exists an exact sequence of U -extension modules of N by M. It is just that the sequence consists of only limited number of nonzero modules. In this paper we investigate the conditions needed by the sequence so that it can be extended into a longer one.

Given any R-module M, in this paper we show that the U-projective resolution of M always exists for any R - submodule U of M. This U-projective resolution is a generalization of ordinary projective resolution, that is when U is the zero submodule, the resolution is just the ordinary projective resolution. This resolution is unique up to U-homotopy and generates the U-extension module as the ordinary projective resolution generates the ordinary extension module. We also show that this U-extension module is unique up to isomorphism for every choice of the U-projective resolution.

Mathematical model of dengue disease with fumigation intervention for mosquito population will discussed in this article. Interaction between human and mosquitoes population will based on SIR-SI host-vector model. Side effect of fumigation where mosquito capable to developed a resistance to fumigation will be accommodated in to the model. Equilibrium points and basic reproductive ratio as the endemic criteria will be shown analytically. Some numerical results are shown to give a back up reasoning for analytical result. We conclude that resistance of mosquitoes to fumigation intervention play an important role in fumigation strategy.