Connected relation explained

In mathematics, a relation on a set is called connected or complete 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

x \in X there is a y \in X so that x l{R} y (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 when for all x, y \in X, $$\backslash text\; x\; \backslash neq\; y\; \backslash text\; x\; \backslash mathrel\; y\; \backslash quad\; \backslash text\; \backslash quad\; y\; \backslash mathrel\; x,$$or, equivalently, when for all x, y \in X, $$x\; \backslash mathrel\; y\; \backslash quad\; \backslash text\; \backslash quad\; y\; \backslash mathrel\; x\; \backslash quad\; \backslash text\; \backslash quad\; x\; =\; y.$$

A relation with the property that for all

x, y \in X, $$x\; \backslash mathrel\; y\; \backslash quad\; \backslash text\; \backslash quad\; y\; \backslash mathrel\; x$$is called .^{[1] [2] [3]}

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.^{[4] [5]} Thus, is used more generally for relations that are connected or strongly connected.^[6] 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,^[7] although this, too, can lead to confusion: The universal relation is also called complete,^[8] and "complete" has several other meanings in order theory.Connected relations are also called ^{[9] [10]} or said to satisfy ^[11] (although the more common definition of trichotomy is stronger in that of the three options

x l{R} y, y l{R} x, 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 and,^[12] and,^[13] and, and,^[14] or and,^[15] 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 ^{\top} ;\overline{R} \subseteq R ^{\top} ;\overline{R} is asymmetric,where U is the universal relation and R ^{\top} is the converse relation of R.

The following are equivalent:

R is connected;\overline{I} \subseteq R \cup R ^{\top} ;\overline{R} \subseteq R ^{\top} \cup I ;\overline{R} is antisymmetric,where \overline{I} is the complementary relation of the identity relation I and R ^{\top} is the converse relation of R.

Introducing progressions, Russell invoked the axiom of connection:

Properties

• The relation^[16]

E of a tournament graph G is always a connected relation on the set of G 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.^[17]
• A connected relation on a set

X cannot be antitransitive, provided X has at least 4 elements.^[18] 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. ^[19] Similarly, all, or all but one, elements of X are in the domain of R.

Notes

Proofs

Notes and References

1. Book: Oxford University Press. 978-0-19-967959-1. Clapham. Christopher. Nicholson. James. The Concise Oxford Dictionary of Mathematics. connected. 2021-04-12. 2014-09-18.
2. Book: Nievergelt, Yves. Springer. 978-1-4939-3223-8. Logic, Mathematics, and Computer Science: Modern Foundations with Practical Applications. 2015-10-13. 182.
3. Book: Causey, Robert L.. Jones & Bartlett Learning. 0-86720-463-X. Logic, Sets, and Recursion. 1994., p. 135
4. Book: Paul R. Halmos. Naive Set Theory. Princeton. Nostrand. 1968. Here: Ch.14. Halmos gives the names of reflexivity, anti-symmetry, and transitivity, but not of connectedness.
5. Book: Patrick Cousot. Methods and Logics for Proving Programs. 841 - 993. 0-444-88074-7. Jan van Leeuwen. Formal Models and Semantics. Elsevier. Handbook of Theoretical Computer Science. B. 1990. Here: Sect.6.3, p.878
6. Book: Springer. 978-3-540-32696-0. Aliprantis. Charalambos D.. Border. Kim C.. Infinite Dimensional Analysis: A Hitchhiker's Guide. 2007-05-02., p. 6
7. Book: Makinson, David. Springer. 978-1-4471-2500-6. Sets, Logic and Maths for Computing. 2012-02-27., p. 50
8. Book: Whitehead. Alfred North. Alfred North Whitehead. Russell. Bertrand. Bertrand Russell. Principia Mathematica. 1910. Cambridge University Press. 1910. Cambridge. English.
9. Book: Wall, Robert E.. Prentice-Hall. Introduction to Mathematical Linguistics. 1974. page 114.
10. Web site: Carl Pollard. Relations and Functions. Ohio State University. 2018-05-28. Page 7.
11. Book: Kunen, Kenneth. College Publications. 978-1-904987-14-7. The Foundations of Mathematics. 2009. p. 24
12. Book: Fishburn, Peter C.. Princeton University Press. 978-1-4008-6833-9. The Theory of Social Choice. 2015-03-08. 72.
13. Book: Roberts, Fred S.. Cambridge University Press. 978-0-521-10243-8. Measurement Theory: Volume 7: With Applications to Decisionmaking, Utility, and the Social Sciences. 2009-03-12. page 29
14. Book: Schmidt. Gunther. Ströhlein. Thomas. 1993. Relations and Graphs: Discrete Mathematics for Computer Scientists. Berlin. Springer. 978-3-642-77970-1. Gunther Schmidt .
15. Book: Springer Science & Business Media. 978-3-642-59830-2. Ganter. Bernhard. Wille. Rudolf. Formal Concept Analysis: Mathematical Foundations. 2012-12-06. p. 86
16. Defined formally by v E w if a graph edge leads from vertex v to vertex w
17. For the direction, both properties follow trivially. - For the direction: when x ≠ y, then x l{R}y \lor y l{R}x follows from connectedness; when x = y, x l{R} y follows from reflexivity.
18. 1806.05036. Jochen Burghardt. Simple Laws about Nonprominent Properties of Binary Relations. Technical Report. Jun 2018. 2018arXiv180605036B. Lemma 8.2, p.8.
19. If x, y \in X \setminus \operatorname{ran}(R), then x l{R} y and y l{R} x are impossible, so x = y follows from connectedness.