A simple example of an order theoretic property for functions comes from analysis where monotone functions are frequently found.
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Büchi arithmetic of base k is the first-order theory of the natural numbers with addition and the function which is defined as the bigger power of k dividing x, named in honor of the Swiss mathematician Julius Richard Büchi.
He revisited Boolean axiomatics in 1933, proving that Boolean algebra required but a single binary operation (denoted below by infix '+') that commutes and associates, and a single unary operation, complementation, denoted by a postfix prime.
This restriction is similar to the restriction to continuous operators in the Kleene fixed-point theorem of order theory.