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If I have a Hamiltonian

$$\mathcal{H} = \prod_j^N Z_j$$

where $j$'s are different sites on a lattice and $Z$'s are Pauli $Z$ operators does that mean that the Hamiltonian can also be written as

$$\mathcal{H} = Z_1 \otimes Z_2 \otimes \cdot \cdot \cdot Z_N$$

and if they are all Pauli operators could it just be

$$\mathcal{H} = Z ^{\otimes N}$$

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Short answer: yes. Long answer: yes.

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