Spectrum topology on hoop algebras

This paper investigates the properties of prime and maximal filters in hoops, introducing two distinct types of prime filters and analyzing their interrelationships. Furthermore, it investigates the existence of maximal filters with a special focus on bounded hoops. Moreover, this paper presents the spectrum of a hoop, and shows that when it has the topology of the spectrum, it forms a compact topological space T0. It is also shown that the maximal spectrum, regarded as a subspace of the spectrum, constitutes a compact T1-topological space. Finally, using the Boolean center of a hoop, the paper establishes that the clopen subsets of the hoop algebra coincide with the set of closed subsets generated by Boolean elements.