Quotients of L-domains

Abstract

It is well known that the image of a bounded complete domain, respectively a continuous lattice T under a surjective map preserving infs of nonempty subsets and directed sups is a bounded complete domain, respectively a continuous lattice and a congruence relation on T is an equivalence relation on T and a subalgebra of T × T . In this paper, we propose some counterexamples to explain that these results do not persist in L -domains. The concepts of strong homomorphisms among L -domains and congruence relation on L -domains are introduced. We prove that the image of an L -domain under a surjective map which preserves infs of nonempty subsets bounded above and directed sups and satisfies an equation is an L -domain. Meanwhile, we obtain the quotients of L -domains. Moreover, we give the Isomorphism Theorem for L -domains.