Groupoid
There are at least two definitions of ``groupoid'' currently in use.
The first type of groupoid is an algebraic structure on a Set with a Binary Operator. The only restriction onthe operator is Closure (i.e., applying the Binary Operator to two elements of a given set 
returns a valuewhich is itself a member of ). Associativity, commutativity, etc., are not required (Rosenfeld 1968, pp. 88-103). Agroupoid can be empty. The numbers of nonisomorphic groupoids of this type having 
elements are 1, 1, 10, 3330, 178981952,... (Sloane's A001329), and the numbers of nonisomorphic and nonantiisomorphic groupoids are 1, 7, 1734, 89521056, ...(Sloane's A001424). An associative groupoid is called a Semigroup.
The second type of groupoid is an algebraic structure first defined by Brandt (1926) and also known as a Virtual Group. A groupoid with base 
is a set 
with mappings 
and 
from onto and a partially defined binaryoperation 
, satisfying the following four conditions:
- 1.
is defined only when 
for certain maps and from onto
with 
and 
- 2. Associativity: If either
or 
is defined, then so is the other and
. - 3. For each
in , there are left and right Identity Elements 
and 
such that
. - 4. Each in has an inverse
for which 
and 
References
Brandt, W. ``Über eine Verallgemeinerung des Gruppengriffes.'' Math. Ann. 96, 360-366, 1926.
Brown, R. ``From Groups to Groupoids: A Brief Survey.'' Bull. London Math. Soc. 19, 113-134, 1987.
Brown, R. Topology: A Geometric Account of General Topology, Homotopy Types, and the Fundamental Groupoid. New York: Halsted Press, 1988.
Higgins, P. J. Notes on Categories and Groupoids. London: Van Nostrand Reinhold, 1971.
Ramsay, A.; Chiaramonte, R.; and Woo, L. ``Groupoid Home Page.'' http://amath-www.colorado.edu:80/math/researchgroups/groupoids/groupoids.shtml.
Rosenfeld, A. An Introduction to Algebraic Structures. New York: Holden-Day, 1968.
Sloane, N. J. A. SequencesA001424 andA001329/M4760in ``An On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.
Weinstein, A. ``Groupoids: Unifying Internal and External Symmetry.'' Not. Amer. Math. Soc. 43, 744-752, 1996.
- Cremona Transformation
- Cribbage
- Criss-Cross Method
- Criterion
- Critical Line
- Critical Point
- Critical Strip
- Crook
- Crookedness
- Cross
- Cross-Cap
- Cross-Correlation
- Cross-Correlation Coefficient
- Cross-Correlation Theorem
- Cross Curve
- Crossed Ladders Problem
- Crossed Trough
- Cross Fractal
- Crossing Number (Graph)
- Crossing Number (Link)
- Cross of Lorraine
- Cross Polytope
- Cross Product
- Cross-Ratio
- Cross-Section