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"Aljabar merupakan suatu ruang vektor yang dilengkapi dengan suatu operator bilinier, yaitu suatu operator yang linier pada masing-masing argumennya. Suatu aljabar dikatakan sebagai aljabar simetris kiri jika asosiator dari sembarang ketiga vektornya simetrik pada kedua argumen pertamanya. Pada skripsi ini dibahas mengenai konstruksi aljabar simetris kiri melalui fungsi linier. Pertama-tama dibahas mengenai konstruksi aljabar secara umum dimana pendefinisian operator bilinier pada aljabar melibatkan fungsi-fungsi linier. Selanjutnya diberikan syarat bagi fungsi linier tersebut sedemikian sehingga aljabar yang telah dikonstruksi merupakan suatu aljabar simetris kiri.

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Algebra is a vector space along with a bilinear operator, that is an operator which is linear on each of its argument. An algebra is called a left symmetric algebra if the associator of any three vectors of it is symmetric on its first two arguments. This skripsi discusses how to construct the left symmetric algebra using linear functions. First, this skripsi discusses how to construct a general algebra on which the bilinear operator defined on the algebra would involve linear functions. Then, some conditions for the linear functions will be given so that the constructed algebra would be a left symmetric algebra."

"The set of real number with addition and multiplication operation is said as field with a notation. At set of is given two binary operations that is and, are defined as follow: for all, and. The structure is called max-plus algebra, which denote as. The main different between and that is there is no invers for all element except the zero elemen. Futhermore are introduced extended of with define and at. In set of are given balance relation, denote, and relation whereas is equivalence relation since its compatible for generate equivalence class and quosien set. Clearly, the elements of set of is equivalence classes and than for all equivalenve class are assosiated with scalar or signed scalar dan in. By using definision of, and balance relation in for design the system of. In this research are performed properties of and than are defined the symmetrized max-plus algebra."