nLab decimal interval coalgebra

Definition

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

$$z:\{m \in \mathbb{N} \vert (m \lt 10)\} \to (I \to I)$$

such that

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

• for all natural numbers $m \in \mathbb{N}$, if $m \lt 10$, then $z(m)(0) = 0$

• for all natural numbers $m \in \mathbb{N}$, if $m \lt 10$, then $z(m)(1) = 1$

• for all natural numbers $m \in \mathbb{N}$, if $m + 1 \lt 10$, then for all elements $a \in I$, it is not true that both $0 \lt z(m+1)(a)$ and $z(m)(a) \lt 1$

Examples

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

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

See also

• decimal rational numbers
• dyadic interval coalgebra
• rational interval coalgebra
• unit interval
• coalgebra