Connected relation
In mathematics, a relation on a set is called connected or total if it relates (or "compares") all pairs of elements of the set in one direction or the other while it is called strongly connected if it relates pairs of elements. As described in the terminology section below, the terminology for these properties is not uniform. This notion of "total" should not be confused with that of a total relation in the sense that for all formula_1 there is a formula_2 so that formula_3 (see serial relation). Connectedness features prominently in the definition of total orders: a  total (or linear) order is a partial order in which any two elements are comparable; that is, the order relation is connected. Similarly, a strict partial order that is connected is a strict total order. A relation is a total order if and only if it is both a partial order and strongly connected. A relation is a strict total order if, and only if, it is a strict partial order and just connected. A strict total order can never be strongly connected (except on an empty domain). Formal definition. A relation formula_4 on a set formula_5 is called when  for all formula_6 formula_7 or, equivalently, when for all formula_6 formula_9 A relation with the property that for all formula_6 formula_11 is called . Terminology. The main use of the notion of connected relation is in the context of orders, where it is used to define total, or linear, orders. In this context, the property is often not specifically named. Rather, total orders are defined as partial orders in which any two elements are comparable. Thus, ' is used more generally for relations that are connected or strongly connected. However, this notion of "total relation" must be distinguished from the property of  being serial, which is also called total. Similarly, connected relations are sometimes called ', although this, too, can lead to confusion: The universal relation is also called complete, and "complete" has several other meanings in order theory. Connected relations are also called or said to satisfy trichotomy (although the more common definition of trichotomy is stronger in that of the three options formula_12 must hold). When the relations considered are not orders, being connected and being strongly connected are importantly different properties. Sources which define both then use pairs of terms such as ' and connected, complete and strongly complete, total and  complete, ' and connex, or connex and , respectively, as alternative names for the notions of connected and strongly connected as defined above. Characterizations. Let formula_4 be a homogeneous relation. The following are equivalent: where formula_18 is the universal relation and formula_19 is the converse relation of formula_20 The following are equivalent: where formula_25 is the complementary relation of the identity relation formula_26 and formula_19 is the converse relation of formula_20 Introducing progressions, Russell invoked the axiom of connection: