Module Structures on Hoops

In this paper, we apply the theory of modules on hoops and introduce two concepts of modules on hoops and provide special examples and interesting results. Both concepts are correct and logical. The first concept is very close to the definition of module in abstract algebra. In this case, we investigate some important results in modules such as sub-modules and quotient structures. But if we want to investigate the relationship between hoop-modules and other modules on logical algebraic structures such as [Formula: see text]-modules and [Formula: see text]-modules, we need to define the second definition of hoop-modules. In this case, we can get that a [Formula: see text]-modules and an [Formula: see text]-module from any hoop-module.