Positive normal identities and bi-order extension

We prove that it is undecidable whether the identity in a free group of any fixed finite rank at least two is a nonempty product of conjugates of given elements, with only positive occurrences of those elements allowed. It is also undecidable whether a finite subset of that group can be made positive in a total order invariant under multiplication on both sides. Both conclusions hold for a fixed finite background together with one varying word, all lying in the commutator subgroup. These give negative answers to the normal identity and bi-order questions stated online in 2025 by Bodart, Ciobanu and Metcalfe, and to the bi-order extension question formulated by Colacito and Metcalfe (2017). The second result also settles their 2019 question about the word problem for free representable lattice-ordered groups: it is undecidable already on two generators.

We further show that every large group, meaning one with a finite-index subgroup mapping onto a free group of rank two, contains a fixed free subgroup that reflects positive normal-semigroup membership, including the number of occurrences of each supplied element. In every fixed finitely generated large group the Normal Identity Problem is undecidable, and no total computable function of the input bounds the number of factors in a positive identity when one exists. No orderability or word-problem hypothesis is imposed on the large ambient group.

GitHub repository: https://github.com/eulogica/positive-normal-identities-paper