Completions of Ordered Algebraic Structures: A Survey

Introduction The ordered field of rationals (Q, ≤, +, ·) is an ordered algebraic structure — i.e. a poset with additional operations. to complete the rationals — that is, to embed the rationals into a structure where every (bounded) subset has a join and meet. This is a very good completion as it preserves all existing joins and meets as well as many algebraic properties. Introduction Ordered algebraic structures play a role in a wide range of areas, including analysis, logic, theoretical computer science, and foundations of physics. In these applications certain infinite joins and meets often play an important role linked to some kind of infinitary or limit process. It is common to seek a completion of such an ordered algebraic structure preserving certain joins and meets and certain aspects of the algebraic structure.