Free semimodules and their examples
August 22, 2024abstract. 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 define, categorically, the property of an algebraic structure being free, and show that this is a universal property. We construct the free semimodule on a set and explore various examples of them appearing in mathematics. We observe that the free semimodule on a set is a universal arrow from that set to the forgetful functor and that there is a functor called the free functor that is the left adjoint to the forgetful functor and which provides an equivalent characterisation of the property of being free. That is, when the object is in the image of the free functor.