On Induction Principles for Partial Orders

Various forms of mathematical induction (induction for natural numbers, transfinite induction, structural induction, well-founded induction, Noetherian induction, etc.) are applicable to domains with some kinds of order. This naturally leads to the questions about the possibility of unification of different inductions and their generalization to wider classes of ordered domains. In the paper we propose a common framework for formulating induction proof principles in various structures and apply it to partially ordered sets. In this framework we propose a fixed induction principle which is indirectly applicable to the class of all posets. In a certain sense, this result provides a solution to the problem of formulating a common generalization of a number of well-known induction principles for discrete and continuous ordered structures.