Semisimplicial Unital Groups

If there is an order-preserving group isomorphism from a directed abelian group G with a finitely generated positive cone G ⁺ onto a simplicial group, then G is called a semisimplicial group. By factoring out the torsion subgroup of a unital group having a finite unit interval, one obtains a semisimplicial unital group. We exhibit a representation for the base-normed space associated with a semisimplicial unital group G as the Banach dual space of a finite dimensional order-unit space that contains G as an additive subgroup. In terms of this representation, we formulate necessary and sufficient conditions for G to be archimedean.