Concurrent Relation

Let  and  be sets, and let  be a relation on . Then  is a concurrent relation if and only if for any finite subset  of , there exists a single element  of  such that if , then . Examples of concurrent relations include the following:

1. The relation  on either the natural numbers, the integers, the rational numbers, or the real numbers.

2. The relation  between elements of an extension  of a field , defined by

3. The containment relation  between open neighborhoods of a given point  of a topological space .

REFERENCES:

Hurd, A. E. and Loeb, P. A. An Introduction to Nonstandard Real Analysis. Orlando, FL: Academic Press, 1985.

Robinson, A. "Germs." In Applications of Model Theory to Algebra, Analysis and Probability (International Sympos., Pasadena, Calif., 1967). New York: Holt, Rinehart and Winston, pp. 138-149, 1969.

Insall, M. "Hyperalgebraic Primitive Elements for Relational Algebraic and Topological Algebraic Models." Studia Logica 57, 409-418, 1996.