4. Standard Discrete Spaces

Summary

Our primary goal in this chapter is to study the discrete positive semigroup \((\N, +)\) with its associated graph \((\N, \le)\). This space is the primary model of discrete time in many applications, including of course, reliability. In addition, we will study some closely related graphs on \(\N\):

• The graph \((\N, \lt)\) which is the reflexive reduction of \((\N, \le)\).
• The covering graph \((\N, \upa)\) of \((\N, \le)\), or a bit more generally, the graph \((\N, \upa^n)\) where \(\upa^n\) is the composition power of \(\upa\) of order \(n \in \N\).
• The reflexive closure \((\N, \rta)\) of \((\N, \upa)\).

There are also corresponding spaces on \(\N_+\) that are isomorphic to the spaces above, and so we do not need to study these separately. In particular,

• The strict positive semigroup \((\N_+, +)\) with associated order \(\lt\). The graph \((\N_+, \lt)\) is isomorphic to \((\N, \lt)\).
• The reflexive closure \((\N_+, \le)\) of \((\N_+, \lt)\), isomorphic to \((\N, \le)\).
• The covering graph \((\N_+, \upa)\) for \((\N_+, \le)\), isomorphic to \((\N, \upa)\).
• The reflexive closure \((\N_+, \rta)\) of \((\N_+, \upa)\), isomorphic to \((\N, \rta)\).

We also consider sub-semigroups of \((\N, +)\), with particular emphasis on numerical semigroups, and we consider graphs induced by \((\N, \le)\) and a special lexicographic sum.

Contents

1. The Standard Space
2. Variations
3. Sub-Semigroups
4. Induced Graphs
5. A Lexicographic Sum