Connected relation

Template:Short description Template:Redirect Template:Stack In mathematics, a relation on a set is called connected or complete or total if it relates (or "compares") all Template:Em pairs of elements of the set in one direction or the other while it is called strongly connected if it relates Template:Em 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 $x\in X$ there is a $y\in X$ so that $xRy$ (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 $R$ on a set $X$ is called Template:Em when for all $x,y\in X,$ $$\text{ if }x\ne y\text{ then }xRy\text{or}yRx,$$ or, equivalently, when for all $x,y\in X,$ $$xRy\text{or}yRx\text{or}x=y.$$

A relation with the property that for all $x,y\in X,$ $$xRy\text{or}yRx$$ is called Template:Em.

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, Template:Em 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 Template:Em, 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 Template:Em or said to satisfy Template:Em (although the more common definition of trichotomy is stronger in that Template:Em of the three options $xRy,yRx,x=y$ 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 Template:Em and Template:Em, Template:Em and Template:Em, Template:Em and Template:Em, Template:Em and Template:Em, or Template:Em and Template:Em, respectively, as alternative names for the notions of connected and strongly connected as defined above.

Characterizations

Let $R$ be a homogeneous relation. The following are equivalent:

• $R$ is strongly connected;
• $U\subseteq R\cup R^{\mathrm{\top }}$;
• $\bar{R}\subseteq R^{\mathrm{\top }}$;
• $\bar{R}$ is asymmetric,

where $U$ is the universal relation and $R^{\mathrm{\top }}$ is the converse relation of $R.$

The following are equivalent:

• $R$ is connected;
• $\bar{I}\subseteq R\cup R^{\mathrm{\top }}$;
• $\bar{R}\subseteq R^{\mathrm{\top }}\cup I$;
• $\bar{R}$ is antisymmetric,

where $\bar{I}$ is the complementary relation of the identity relation $I$ and $R^{\mathrm{\top }}$ is the converse relation of $R.$

Introducing progressions, Russell invoked the axiom of connection: Template:Blockquote

Properties

• The Template:Em relation $E$ of a tournament graph $G$ is always a connected relation on the set of $G$Template:'s vertices.
• If a strongly connected relation is symmetric, it is the universal relation.
• A relation is strongly connected if, and only if, it is connected and reflexive.
• A connected relation on a set $X$ cannot be antitransitive, provided $X$ has at least 4 elements. On a 3-element set $\{a,b,c\},$ for example, the relation $\{(a,b),(b,c),(c,a)\}$ has both properties.
• If $R$ is a connected relation on $X,$ then all, or all but one, elements of $X$ are in the range of $R.$ Similarly, all, or all but one, elements of $X$ are in the domain of $R.$

Notes

Proofs