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Concept# Serial relation

Summary

In set theory a serial relation is a homogeneous relation expressing the connection of an element of a sequence to the following element. The successor function used by Peano to define natural numbers is the prototype for a serial relation.
Bertrand Russell used serial relations in The Principles of Mathematics (1903) as he explored the foundations of order theory and its applications. The term serial relation was also used by B. A. Bernstein for an article showing that particular common axioms in order theory are nearly incompatible: connectedness, irreflexivity, and transitivity.
A serial relation R is an endorelation on a set U. As stated by Russell, where the universal and existential quantifiers refer to U. In contemporary language of relations, this property defines a total relation. But a total relation may be heterogeneous. Serial relations are of historic interest.
For a relation R, let {y: xRy } denote the "successor neighborhood" of x. A serial relation can be equivalently characterized as a relation for which every element has a non-empty successor neighborhood. Similarly, an inverse serial relation is a relation in which every element has non-empty "predecessor neighborhood".
In normal modal logic, the extension of fundamental axiom set K by the serial property results in axiom set D.
Relations are used to develop series in The Principles of Mathematics. The prototype is Peano's successor function as a one-one relation on the natural numbers. Russell's series may be finite or generated by a relation giving cyclic order. In that case, the point-pair separation relation is used for description. To define a progression, he requires the generating relation to be a connected relation. Then ordinal numbers are derived from progressions, the finite ones are finite ordinals. Distinguishing open and closed series results in four total orders: finite, one end, no end and open, and no end and closed.
Contrary to other writers, Russell admits negative ordinals.

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Homogeneous relation

In mathematics, a homogeneous relation (also called endorelation) on a set X is a binary relation between X and itself, i.e. it is a subset of the Cartesian product X × X. This is commonly phrased as "a relation on X" or "a (binary) relation over X". An example of a homogeneous relation is the relation of kinship, where the relation is between people. Common types of endorelations include orders, graphs, and equivalences. Specialized studies of order theory and graph theory have developed understanding of endorelations.

Binary relation

In mathematics, a binary relation associates elements of one set, called the domain, with elements of another set, called the codomain. A binary relation over sets X and Y is a new set of ordered pairs (x, y) consisting of elements x in X and y in Y. It is a generalization of the more widely understood idea of a unary function. It encodes the common concept of relation: an element x is related to an element y, if and only if the pair (x, y) belongs to the set of ordered pairs that defines the binary relation.

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