[This puzzle is heavily based on one from video game Zero Time Dilemma.]

I have one of those weird individualist friends--to give you an idea, he spells his name H-E-N-🔟-R-Y. ...The 🔟 is silent, y'see.

Anyway, he's easy enough to talk to but, like most people around this generation, he insists on texting. And the thing is, he uses his own number system--it's base-13 and, furthermore, is expressed entirely in emojis. It's always difficult to settle on a time to meet when he says he's available at 😏:🙃🙃.

I've managed to figure out these rules.

😳 is greater than 😔, which is greater than 😎.

😑 times 😆 equals 😋.

😑 times 🙄 times 😑 equals 😏.

😬 equals 😑 times 😑.

Anything multiplied by 🙃 is 🙃.

🙄 times 🙄 equals 😂.

😑 times 😬 is 😠.

😑 times 🙄 equals 😘.

From here, can you determine what emojis correspond to what numbers, 0-C (zero through 12, base ten)?

🙃 = 0 (because 🙃 * x = 🙃)

Because

😬 equals 😑 times 😑 we know
😑 = 2 or 3
😬 = 4 or 9

But

😑 times 😬 is 😠
so 😬 must be 4 because 9 x something > 1 would be too large
This means 😑 = 2

So

2 times 🙄 times 2 equals 😏. Which means 🙄 can be 1 or 3
And 😏 is 4 or 12. But 4 is 😬, so 😏 is 12 and 🙄 is 3

Thus

🙄 times 🙄 equals 😂. Means
😂 = 9

And

😑 times 😬 is 😠 means 😠 is 8
😑 times 🙄 equals 😘. Means 😘 = 6

Because

2 times 😆 equals 😋.
😆 must be 5 based on what’s left and 😋 = 10

And

😳 is greater than 😔, which is greater than 😎 means
😎 = 1, 😔 = 7 and 😳 = 11 because they are the only ones left

So

🙃 = 0
😎 = 1
😑 = 2
🙄 = 3
😬 = 4
😆 = 5
😘 = 6
😔 = 7
😠 = 8
😂 = 9
😋 = 10
😳 = 11
😏 is 12

• Well done! I think you solved this faster than I did in the game (which had the advantage of better-organized clues and prime factoring in its number construction). – Exal Dec 15 '18 at 23:14