Frame congruences via subkernels

Abstract

To study the quotient of algebras, like frames, whose algebraic structures are determined by a partial order, it is often more common to think about sub-kernels of homomorphisms between such algebras. So, in this paper, we first introduce the concept of a pre-congruence on a frame and then characterize them as the sub-kernels of the frame homomorphisms. Second, we characterize the frame congruences as the intersection of a pre-congruence and its inverse. Finally, we prove the decomposition and isomorphism theorems for frame homomorphisms.