LATTICE EMBEDDINGS AND PUNCTUAL LINEAR ORDERS
M. V. Dorzhieva, Ellen Hammatt · Journal of Symbolic Logic · 2025
Abstract We investigate the primitive recursive content of linear orders. We prove that the punctual degrees of rigid linear orders, the order of the integers $\mathbb {Z}$ , and the order of the rationals $\mathbb {Q}$ embed the diamond (preserving supremum and infimum). In the cases of rigid orders and the order $\mathbb {Z}$ , we further extend the result to embed the atomless Boolean algebra; we leave the case of $\mathbb {Q}$ as an open problem. We also show that our results for the rigid orders, in fact, work for orders having a computable infinite invariant rigid sub-order.