Normal-semigroup generators for the bi-order cones of Thompson's group F′

The commutator subgroup F′ of Thompson's group has four bi-invariant total orders, determined by the slope at the first or last point of support. We prove that the positive cone of each order is generated by two elements as a normal semigroup: every positive element is a nonempty product of conjugates of those two elements. An explicit pair f, g works for all four cones, using the four choices of signs (f, g), (f⁻¹, g⁻¹), (f⁻¹, g), and (f, g⁻¹). This answers the finite-generation question posed by Navas and Rivas in 2008, including their stronger requirement that the same pair yield both prescribed partitions of F′\{1}. Two generators are necessary for each cone.

GitHub repository: eulogica/normal-semigroup-biorder-cones-fprime