sentences:  For some the motivation has been an interest in the conceptual boundaries of lattice theory; for others it was a search for noncommutative forms of logic and Boolean algebra; and for others it has been the behavior of idempotents in rings.A "noncommutative lattice", generally speaking, is an algebra formula_11 where formula_6 and formula_5 are associative, idempotent binary operations connected by absorption identities guaranteeing that formula_6 in some way dualizes formula_5.The precise identities chosen depends upon the underlying motivation, with differing choices producing distinct varieties of algebras.