nLab rational interval coalgebra

Definition

A rational interval coalgebra is a set $I$ with a linear order $\lt$, elements $0\in I$, $1 \in I$, and a partial function from $\mathbb{N} \times \mathbb{N}$ to the set of endofunctions in $I$, $I \to I$

$$z:\{(m,n) \in \mathbb{N} \times \mathbb{N} \vert isPrime(n) \wedge (m \lt n)\} \to (I \to I)$$

such that

• for all elements $a \in I$, $0 \lt a$ or $a \lt 1$

• for all natural numbers $n \in \mathbb{N}$ and $m \in \mathbb{N}$, $n$ is a prime number, and if $m \lt n$, then $z(n,m)(0) = 0$

• for all natural numbers $n \in \mathbb{N}$ and $m \in \mathbb{N}$, $n$ is a prime number, and if $m \lt n$, then $z(n,m)(1) = 1$

• for all natural numbers $n \in \mathbb{N}$ and $m \in \mathbb{N}$, $n$ is a prime number, and if $m + 1 \lt n$, then for all elements $a \in I$, it is not true that both $0 \lt z(n,m+1)(a)$ and $z(n,m)(a) \lt 1$.

Examples

• The initial rational interval coalgebra is the unit interval on the rational numbers

• The terminal rational interval coalgebra is the unit interval on the Dedekind real numbers

See also

• rational number
• dyadic interval coalgebra
• decimal interval coalgebra
• unit interval
• coalgebra