Dilation of time scales

Let $\mathbb{T}_1$ and $\mathbb{T}_2$ be time scales. Their dilation, $\mathbb{T}_1 \oplus \mathbb{T}_2$, is a time scale defined by $$\mathbb{T}_1 \oplus \mathbb{T}_2=\{t_1+t_2 \colon t_1 \in \mathbb{T}_1, t_2 \in \mathbb{T}_2\}.$$

Examples

• $\{0,1\} \oplus \{4,5,10\} = \{4,5,6,10,11\}$
• $\{0,2\} \oplus [0,1] = [0,1] \cup [2,3]$
• $2\mathbb{Z} \oplus \mathbb{Z}=\mathbb{Z}$

References

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