Some Results on Filters and Congruence Relations of Hoops

This manuscript presents some results on the conditions and characterizations of filters, congruence relations and homomorphisms in algebraic hoops. The paper investigates the structural properties of filters that are generated by a set in various ways and establishes several descriptions for such filters. It is demonstrated in this paper that the class of filters in hoops forms an algebraic lattice. This finding contributes to our understanding of the structural properties of filters and their relationship within hoops. Moreover, this manuscript explores congruence relations in hoops, revealing a fascinating connection between the lattice of filters and the lattice of congruences. In particular, it is shown that the lattice of filters is in one-to-one correspondence with the lattice of congruences in hoops, confirming that the variety of hoops is an ideal determined. Additionally, the manuscript delves into the properties of hoop homomorphisms in relation to congruences and filters, as well as to quotient structures. This paper presents several homomorphism and correspondence theorems for hoops.