Free rieg elements as games

Question

Let $F_1$ be the free rieg on one generator $x$. Can its elements be interpreted as combinatorial games? Can the unique rieg homomorphism

$$F_1\longrightarrow\mathbb N,\qquad x\longmapsto0$$

be interpreted in terms of those games?

What an answer must provide

Specify the games and their equality, interpret $0,1,+,\times$, and exponentiation, and show that the rieg equations hold. A full model of $F_1$ must also identify when two game expressions represent the same element; a map out of $F_1$ alone need not be injective.

Next step

Give a concrete candidate for exponentiation of games, then test the exponent laws and the evaluation $x\mapsto0$. The existing recursive-coalgebra model of impartial games does not by itself supply these rieg operations.

Sources: original problem 7.0.1; Classical vs quantum computation.

@misc{sumire:2609.00007:3:v4,
  author = {hora},
  title = {{Free rieg elements as games}},
  year = {2026},
  note = {SUMIRE:2609.00007/3, v4},
  url = {https://sumire.aiformath.jp/p/2609.00007/n/3/v/4}
}