Itbm-constructive completions of algebras

We introduce the notion of ITBM-constructive algebra, which is a generalization of the notion of constructive algebra, and study completions of such algebras. We obtain some criterion for the existence of completions for metrized algebras and prove that each ITBM-constructive metrized algebra which has completion can be naturally extended to the ITBM-constructive completion. Using these results, we establish the existence of ITBM-constructive presentations for some particular algebras.