Reactive Systems: Modelling, Specification and Verification - Cs.ioc.ee

86 CHAPTER 4. THEORY OF FIXED POINTS
poset) is a pair (D, ⊑), where D is a set, and ⊑ is a binary relation over D (i.e., a
subset of D × D) such that:
• ⊑ is reflexive, i.e., d ⊑ d for all d ∈ D;
• ⊑ is antisymmetric, i.e., d ⊑ e and e ⊑ d imply d = e for all d, e ∈ D;
• ⊑ is transitive, i.e., d ⊑ e ⊑ d ′ implies d ⊑ d ′ for all d, d ′ , e ∈ D.
We moreover say that (D, ⊑) is a totally ordered set if, for all d, e ∈ D, either
d ⊑ e or e ⊑ d holds.
Example 4.1 The following are examples of posets.
• (N, ≤), where N denotes the set of natural numbers, and ≤ stands for the
standard ordering over N.
• (R, ≤), where R denotes the set of real numbers, and ≤ stands for the standard
ordering over R.
• (A ∗ , ≤), where A ∗ is the set of strings over alphabet A, and ≤ denotes the
prefix ordering between strings, i.e., for all s, t ∈ A ∗ , s ≤ t iff there exists
w ∈ A ∗ such that sw = t. (Check that this is indeed a poset!)
• Let (A, ≤) be a finite totally ordered set. Then (A ∗ , ≺), the set of strings
in A ∗ ordered lexicographically, is a poset. Recall that, for all s, t ∈ A ∗ ,
the relation s ≺ t holds with respect to the lexicographic order if one of the
following conditions apply:
1. the length of s is smaller than that of t;
2. s and t have equal length, and either s = ε or there are strings u, v, z ∈
A ∗ and letters a, b ∈ A such that s = uav, t = ubz and a ≤ b.
• Let (D, ⊑) be a poset and S be a set. Then the collection of functions from
S to D is a poset when equipped with the ordering relation defined thus:
f ⊑ g iff f(s) ⊑ g(s), for each s ∈ S .
We encourage you to think of other examples of posets you are familiar with.
Exercise 4.1 Convince yourselves that the structures mentioned in the above example
are indeed posets. Which of the above posets is a totally ordered set?