In mathematics, a well-order (or well-ordering or well-order relation) on a set S is a total ordering on S with the property that every non-empty subset...

the well-ordering theorem, also known as Zermelo's theorem, states that every set can be well-ordered. A set X is well-ordered by a strict total order if...

In mathematics, the well-ordering principle states that every non-empty subset of nonnegative integers contains a least element. In other words, the set...

In mathematics, specifically order theory, a well-quasi-ordering or wqo on a set X {\displaystyle X} is a quasi-ordering of X {\displaystyle X} for which...

In order theory, a partial order is called well-founded if the corresponding strict order is a well-founded relation. If the order is a total order then...

lexicographic or lexicographical order (also known as lexical order, or dictionary order) is a generalization of the alphabetical order of the dictionaries to sequences...

mathematics, a total order or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation ≤...

∗ {\displaystyle \sigma ^{*}} . The order type of a well-ordered set X is sometimes expressed as ord(X). The order type of the integers and rationals is...

or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory. In calculus,...

Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing...

In the mathematical field of order theory, an order isomorphism is a special kind of monotone function that constitutes a suitable notion of isomorphism...

linear order such that every non-empty subset has a least element is called a well-order. The axiom of choice implies that every set can be well-ordered...

transfinite induction: First, well-order the real numbers (this is where the axiom of choice enters via the well-ordering theorem), giving a sequence ⟨...

In mathematical order theory, an ideal is a special subset of a partially ordered set (poset). Although this term historically was derived from the notion...

In the mathematical area of order theory, completeness properties assert the existence of certain infima or suprema of a given partially ordered set (poset)...

is called orderable or linearly orderable if there exists a total order on its elements such that the order topology induced by that order and the given...

New World Order is a non-fiction book written by H.G. Wells and was published by Secker & Warburg in January 1940. In The New World Order, Wells proposed...

In mathematics, especially order theory, a partial order on a set is an arrangement such that, for certain pairs of elements, one precedes the other....

natural numbers. It begins by defining well-ordered sets. Ordinal numbers are then introduced as the order types of well-ordered sets. Cantor then defines...

order relations on the set of monomials that are not well-orders. In the case of finitely many variables, well-ordering of a monomial order is equivalent...

In mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive and transitive. The name preorder is meant...

{\displaystyle L} . It is well known that the axiom of choice is equivalent to the ability to well-order every set. Being able to well-order the proper class V...

this preorder is even a partial order (called the specialization order). On the other hand, for T1 spaces the order becomes trivial and is of little...

In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. Basic examples...

In mathematics, especially order theory, a weak ordering is a mathematical formalization of the intuitive notion of a ranking of a set, some of whose...

incomparabilities) Well-orders, total orders in which every non-empty subset has a least element Well-quasi-orderings, a class of preorders generalizing the well-orders...

The Most Excellent Order of the British Empire is a British order of chivalry, rewarding contributions to the arts and sciences, work with charitable and...

other hand, it might well be that two (necessarily infinite) posets are mutually order-embeddable into each other without being order-isomorphic. An example...

B} , respectively, the product order (also called the coordinatewise order or componentwise order) is a partial ordering ≤ {\displaystyle \leq } on the...

The Order of Australia is an Australian honour that recognises Australian citizens and other persons for outstanding achievement and service. It was established...