abstract. This project covers free semimodules and their examples. We define various algebraic structures via $\Omega$-algebras (from universal algebra) with an emphasis on semimodules. We continue to …

Consider a vector space $V$, when $X$ is a basis of $V$, we say that $V$ is a free vector space on $X$ 1. For any vector space $V$ and $W$ and a basis $X$ of $V$, if there is a map $f:X\to W$ then …

An algebraic category is defined to be any full subcategory of $\textbf{Alg}(\Omega)$, the catgeory of all $\Omega$-algebras. We can define various algebraic structures as algebraic categories. For …