By Jean-Éric Pin

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2 Orders on words The relation “being a prefix of” is an order relation on A∗ , called the prefix order. which is partial if A contains at least two letters. There are other interesting orders on the free monoid. We describe two of them, the lexicographic order and the shortlex order. Let A be an alphabet and let be a total order on A. The lexicographic order induced by is the total order on A∗ used in a dictionary. Formally, it is the order lex on A∗ defined by u lex v if and only if u is a prefix of v or u = pau′ and v = pbv ′ for some p ∈ A∗ , a, b ∈ A with a < b.

Sn of n elements of S, there exists an index i ∈ {1, . . , n} and an idempotent e ∈ S such that s1 · · · si e = s1 · · · si . Proof. Consider the sequence s1 , s1 s2 , . . , s1 · · · sn . If these elements are all distinct, the sequence exhausts the elements of S and one of them, say s1 · · · si , is idempotent. The result is thus clear in this case. Otherwise, two elements of the sequence are equal, say s1 · · · si and s1 · · · sj with i < j. Then we have s1 · · · si = s1 · · · si (si+1 · · · sj ) = s1 · · · si (si+1 · · · sj )ω where ω is the exponent of S.

Diekert and Gastin) Let M be a monoid and let m be an element of M . (1) Show that the operation ◦ defined on the set mM ∩ M m by xm ◦ my = xmy is well defined. (2) Show that (mM ∩ M m, ◦) is a monoid with identity m which divides M . Section 4 34 CHAPTER II. SEMIGROUPS AND BEYOND 5. Show that, if n table below 2, Tn is generated by the three functions a, b, and c of the 1 a b c 1 2 2 1 2 3 1 2 3 4 3 3 ··· ··· ··· ··· n−1 n n−1 n−1 n 1 n 1 An element s of Tn is a function from {1, . . , n} into itself.