Anne C. Morel's contributions to the theories of complete lattices and order types

This paper presents the main contributions of Anne C. Morel regarding the theories of complete lattices and order types. In studying order types, Morel initially investigated their arithmetic, focusing on finite multiplication. In fact, she wondered what the left cancelling order types were, i.e., for which order types α from α∙β=α∙γ follows β=γ. The mathematician deduced that the following are equivalent: (a) α is a left cancelling type; (b) α≠α+α∙β+α ∀β ; (c) α≠α∙β ∀β≠1. From this, it can be derived that every scattered type is a left cancelling one. Morel subsequently looked for the solutions of the equation ξ^n=α and she demonstrated that if α is scattered then ξ^n=α has at most one solution otherwise there exist 2^(ℵ_0 ) distinct numerable non-scattered order types α such that ξ^n=α has exactly m solutions, with m a finite cardinal or m=ℵ_0 or m=2^(ℵ_0 ) and n≥2. Regarding complete lattices, she started from a theorem proved by Tarski which states that every complete lattice (L,≤) has the property that every increasing function f:L→L has a fixpoint. Morel proved the sufficiency of this theorem: she showed that every incomplete lattice has an increasing function with no fixpoints. Therefore, thanks to her contribution, a characterization of complete lattices was obtained.