In this chapter, we study homomorphisms between lattices. Many of the concepts and results are analogous to those typically studied in group and ring theory. For example, the concept of lattice homomorphism is analogous to that of group homomorphism. These concepts have natural applications in the theory of distributed systems. In particular, we show applications in predicate detection.

We begin by defining functions that preserve the order structure.

The concept of preserving order can be extended to preserving meet and join operations.

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