Foundations of Temperature Theory

Temperature of combinatorial games have been long studied since when Conway established the modern combinatorial game theory. Although several variations of the concepts have been developed, there seems yet no presentation of the foundations of temperature theory (including a thorough proof of the existence of temperature, for example) which is rigorous and easy to understand. In this article, we will give precise proofs to the fundamental claims on temperature, namely, the existence, order-preservation, and homomorphism.