By N. Benvenuto, et al.,
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Extra resources for Algorithms for Comm. Systs and Their Applns
We have ∀i, ∩i Xi ⊆ Xi , and monotonicity of f (Prop. 35) gives ∀i, f (∩i Xi ) ⊆ f (Xi ). This entails f (∩i Xi ) ⊆ ∩i f (Xi ). By monotonicity, ∀i, Xi ⊆ ∪i Xi ⇒ f (Xi ) ⊆ f (∪i Xi ). Hence, ∪i f (Xi ) ⊆ f (∪i Xi ). Stronger properties are interesting, where inclusions are replaced by equalities. Intersection is equivalent to conjunction, and union is equivalent to disjunction, whence the generic term “junctivity”. Such junctivity properties are useful when using various ﬁxpoint theorems. All of them are further discussed in [284, 93].
Finally, nondeterminism and probabilities can be related, like in the semantics of programs.
Therefore, we introduce an iteration scheme based on transﬁnite ordinal numbers, which generalizes classical iteration schemes. 5. Before introducing this chapter, some useful notational conventions are ﬁrst mentioned. They concern sequences of arbitrary spaces, or words of formal languages. 1 (Sequences, words). Let X is be an arbitrary space (resp. an alphabet). Then X ≤n is the set of sequences (resp. words) of length smaller than n, X n is the set of sequences of length n, X ∗ is the set of ﬁnite sequences of X (including the empty sequence ε), X ω is the set of inﬁnite sequences, X ∞ = Σ ∗ ∪ Σ ω , X n>ω is the set of sequences longer than ω, and X O = X ∗ ∪ X ω ∪ X n>ω .