Szpilrajn extension theorem

The set of Marczewski - Szpilrajn, sometimes only set of Szpilrajn, named after the Polish mathematician Edward Marczewski, who bore the name Szpilrajn to 1940, is a mathematical theorem from the theory of order. He says that any partial order can be extended to a linear order.

A partial order is a non- empty set together with a two - place relation, so that

• For all elements, for the non-existence of order is ( irreflexivity )
• Off and follows ( transitivity ) for all elements.

The partial order is called linear if its elements are either identical or are in an order relation.

The usual arrangement < on the set of real numbers is a linear order. If we define the order

It is a partial order, which is not linear.

• Set of Marczewski - Szpilrajn: Every partial order can be extended to a linear order.

Specifically, this means that there < are on each partially ordered set is a linear order so that always follows. In the above example about the lexicographic order is a linear order, which continues.

It shows this set initially by induction on finite sets and leads the general case back by the compactness theorem to the case of finite sets, as in the below textbook by Philip Roth painter is executed.