Abstract
This paper presents two axiomatic description of infinite root systems in a base free way. In 1982, Moody and Yokonuma proposed a set of axioms for infinite root systems. These axioms are not general enough to capture all objects that one would intuitively recognize as root systems. The Moody and Yokonuma axioms are expanded to obta n the geometric root system axioms, which capture objects missed by the Moody and Yokonuma axioms. As well, a new set of axioms, rational root systems, is presented. Unlike other axiom systems for root systems, rational root systems are independent of any assumptions about the underlying field. Another system of axioms, root data, has been developed by Moody and Pianzola. The three axiom systems are shown to be 'equivalent.
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