Definition:Ordering on Natural Numbers/1-Based
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Definition
Let $\N_{>0}$ be the axiomatised $1$-based natural numbers.
The strict ordering of $\N_{>0}$, denoted $<$, is defined as follows:
- $\forall a, b \in \N_{>0}: a < b \iff \exists c \in \N_{>0}: a + c = b$
The (weak) ordering of $\N_{>0}$, denoted $\le$, is defined as:
- $\forall a, b \in \N_{>0}: a \le b \iff a = b \lor a < b$
Also see
- Ordering on $1$-Based Natural Numbers is Total Ordering demonstrating that this relation $<$ is in fact a (strict) total ordering.
Sources
- 1964: W.E. Deskins: Abstract Algebra ... (previous) ... (next): $2.2$: Definition $2.2$