Question: Let [n] = {1, 2, . . . , n}. Let Ln be the set of all binary relations on [n] that contain only those

Let [n] = {1, 2, . . . , n}. Let Ln be the set of all binary relations on [n] that contain only those pairs (i, j), where i j. (Examples: S = {(1, 2), (2, 3), (3, 3)} is in R3, but T ={(1, 3), (3, 1)} is not due to the pair (3, 1).) Let An be the Kleene algebra containing the set of all relations on [n].

(a) Show that Rn is a sub-Kleene algebra of Ln. (What you need to show, is that Ln contains 0 and 1 and that non of the three operations +, and take you out of Ln.)

(b) Can you find another proper sub-Kleene algebra of An besides Ln?

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