Brian Cohen wrote:

>An aside on the topic of partial orders in currency: perhaps we can take inspiration to what humans did in the past when (totally-ordered) bullion money was scarce but still needed to do commerce: we often relied on recording transactions with others through tally sticks and other forms of credit. You could even make it effectively tamper-proof by recording transactions in which two halves of a stick, forging the exact locations of notches natural grain of the split stick would have been practically impossible. Debt in this form could then be traded as currency and someone to try to collect on that debt.

>This is partially ordered because a mark on one set of sticks is not necessarily fungible or comparable with another: one person's debt might not be perceived as being able to pay back their debt, so in exchange, it might not be perceived worth the nominal value.

What you're describing is Double-Entry Bookkeeping. David Ellerman (the same author from the [Partition Logic](https://arxiv.org/abs/0902.1950) paper) also made

a paper giving a mathematical treatment of double-entry bookkeeping: [On Double-Entry Bookkeeping: The Mathematical Treatment](https://arxiv.org/abs/1407.1898).

Fredrick Eisele wrote:

>...what do we do when x<0?

Why not simply map \\(x \in \mathbb{R} \\) to \\( 0 \in \mathbb{N} \\)?

>An aside on the topic of partial orders in currency: perhaps we can take inspiration to what humans did in the past when (totally-ordered) bullion money was scarce but still needed to do commerce: we often relied on recording transactions with others through tally sticks and other forms of credit. You could even make it effectively tamper-proof by recording transactions in which two halves of a stick, forging the exact locations of notches natural grain of the split stick would have been practically impossible. Debt in this form could then be traded as currency and someone to try to collect on that debt.

>This is partially ordered because a mark on one set of sticks is not necessarily fungible or comparable with another: one person's debt might not be perceived as being able to pay back their debt, so in exchange, it might not be perceived worth the nominal value.

What you're describing is Double-Entry Bookkeeping. David Ellerman (the same author from the [Partition Logic](https://arxiv.org/abs/0902.1950) paper) also made

a paper giving a mathematical treatment of double-entry bookkeeping: [On Double-Entry Bookkeeping: The Mathematical Treatment](https://arxiv.org/abs/1407.1898).

Fredrick Eisele wrote:

>...what do we do when x<0?

Why not simply map \\(x \in \mathbb{R} \\) to \\( 0 \in \mathbb{N} \\)?