In abstract algebra, a central groupoid is an algebraic structure defined by a binary operation ${\displaystyle \cdot }$ on a set of elements that satisfies the equation $${\displaystyle (a\cdot b)\cdot (b\cdot c)=b.}$$ These structures have bijections to the central digraphs, directed graphs that have exactly one two-edge path between every two vertices, and (for finite central groupoids) to the (0,1)-matrices whose squares are the all-ones matrices.

As an example, the operation ${\displaystyle \cdot }$ on points in the Euclidean plane, defined by recombining their Cartesian coordinates as $${\displaystyle (x_{1},y_{1})\cdot (x_{2},y_{2})=(y_{1},x_{2})}$$ is a central groupoid. The same type of recombination defines a central groupoid over the ordered pairs of elements from any set, called a natural central groupoid.

As an algebraic structure with a single binary operation, a central groupoid is a special kind of magma or groupoid. Because central groupoids are defined by an equational identity, they form a variety of algebras in which the free objects are called free central groupoids. Free central groupoids are infinite, and have no idempotent elements. Finite central groupoids, including the natural central groupoids over finite sets, always have a square number of elements, whose square root is the number of idempotent elements.

A central groupoid consists of a set of elements and a binary operation ${\displaystyle \cdot }$ on this set that satisfies the equation $${\displaystyle (a\cdot b)\cdot (b\cdot c)=b}$$ for all elements ${\displaystyle a}$, ${\displaystyle b}$, and ${\displaystyle c}$.

Central groupoids can be defined equivalently in terms of central digraphs. These are directed graphs in which, each ordered pair of vertices (not necessarily distinct) form the start and end vertex of a three-vertex directed walk. That is, for each ${\displaystyle u}$ and ${\displaystyle v}$ there must exist a unique vertex ${\displaystyle w}$ such that ${\displaystyle u\to w}$ and ${\displaystyle w\to v}$ are directed edges. From any central digraph, one can define a central groupoid in which ${\displaystyle u\cdot v=w}$ for each directed path ${\displaystyle u\to w\to v}$. Conversely, for any central groupoid we can define a central digraph by letting the set of vertices be the elements of the groupoid, and saying there is an edge ${\displaystyle u\to w}$ whenever there exists ${\displaystyle v}$ with ${\displaystyle u\cdot v=w}$.

A third equivalent definition of central groupoids involves (0,1)-matrices ${\displaystyle M}$ with the property that ${\displaystyle M^{2}}$ is a matrix of ones. These are exactly the directed adjacency matrices of the finite graphs that define finite central groupoids.

Every finite central groupoid has a square number of elements. If the number of elements is ${\displaystyle k^{2}}$, then there are exactly ${\displaystyle k}$ idempotent elements (elements ${\displaystyle i}$ with the property that ${\displaystyle i\cdot i=i}$). In the corresponding central digraph, each idempotent vertex has a self-loop. The remaining vertices each belong to a unique 2-cycle. In the matrix view of central groupoids, the idempotent elements form the 1s on the main diagonal of a matrix representing the groupoid. Each row and column of the matrix also contains exactly ${\displaystyle k}$ 1s. The spectrum of the matrix is ${\displaystyle k,0,0,\dots ,0}$. The rank ${\displaystyle r}$ of the matrix can be any number in the range ${\displaystyle k\leq r\leq \lfloor (k+1)^{2}/2\rfloor }$.

The numbers of central groupoids on ${\displaystyle k^{2}}$ labeled elements, or equivalently, (0,1)-matrices of dimension ${\displaystyle k^{2}\times k^{2}}$ whose square is the all-ones matrix, for ${\displaystyle k=1,2,3}$, are