Discrete Algebraic Methods by Volker Diekert Manfred Kufleitner Gerhard Rosenberger Ulrich Hertrampf

Author:Volker Diekert,Manfred Kufleitner,Gerhard Rosenberger,Ulrich Hertrampf
Language: eng
Format: epub
Publisher: De Gruyter
Published: 2016-07-15T00:00:00+00:00

6.3. Determine a nonfree monoid, in which for all elements p, q, x, y with pq = xy there is an element u such that either p = xu, y = uq or x = pu, q = uy is satisfied (compare with Exercise 6.2).

6.4. Let < be a linear order on Σ. We define the lexicographic order ≺ on Σ∗ by u ≺ υ if u is a proper prefix of υ or u = ras and υ = rbt with r, s, t ∈ Σ∗, a, b ∈ Σ and a < b. Show:

(a)∀w ∈ Σ∗ : u ≺ υ ⇔ wu ≺ wυ

(b)If u is not a prefix of υ, then ∀w, z ∈ Σ∗ : u ≺ υ ⇒ uw ≺ υz.

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