# Bitcoin’s mathematical definition (1)

We want to formulate the data behind Bitcoin more scientifically since it got also attention from the science area. We start with some simple and basic defintions:

• $$(B, <)$$: Bounded lattice-ordered set of all blocks
• $$(h, B_{h}, ts_{h}) \in B: h \in \mathbb{N}_0, ts_{h} \in \mathbb{N}_0$$: $$h$$th element of all blocks where $$B_{h}$$ is the (unordered) set of transactions in block with height $$h$$ and $$ts_{h}$$ the associated timestamp
• $latext t_{x} \in B_{h}$: Transaction $$t$$ with id $$x$$ in $$B_{h}$$
• $$A$$: Set of all addresses
• $$(S_{t_{x}}^{in}, <)$$: Bounded lattice-ordered set of all inputs of $$t_{x}$$
• $$(S_{t_{x}}^{out}, <)$$: Bounded lattice-ordered set of all outputs of $$t_{x}$$
• $$(i, a_{i}, v_{i}) \in S_{t_{x}}^{in}: i \in \mathbb{N}_0, a_{i} \in A, v_{i} \in \mathbb{N}_0^{+}$$: $$i$$th element of the inputs
• $$(j, a_{j}, v_{j}) \in S_{t_{x}}^{out}: j \in \mathbb{N}_0, a_{j} \in A, v_{j} \in \mathbb{N}_0^{+}$$: $$j$$th element of the outputs with the condition: $$(i, a_{i}, v_{i}) \in S_{t_{x}}^{in} . \exists! (j, a_{j}, v_{j}) \in S_{t_{y}}^{out} \land a_{i} = a_{j} \land v_{i} = v_{j}$$