Monoids, Boolean Algebras, Materially Ordered Sets

In this paper, the interplay between certain mathematical structures is elucidated. First, it is shown that there is a one-to-one correspondence between bounded half-lattices and commutative idempotent monoids (c.i.-monoids). Adding certain additional structural ingredients and axioms, such c.i.-momoids become Boolean algebras. There is a non-trivial one-to-one correspondence between these and what we call materially ordered sets, which are half -lattices that satisfy certain additional axioms. Such materially ordered sets can serve as mathematical models for certain physical systems. The correspondence between materially ordered sets and Boolean algebras can be used to show, for example, that the law of action and reaction (Newton’s third law) is not an independent axiom but a consequence of fundamental balance laws.