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I just remember an example by myself.

Let $C$ be the following algebraic category: Objects are nonempty sets $G$ together with a binary operation $/ : G \times G \to G$, such that for all $x,y,z \in G$, we have

$x/((((x/x)/y)/(((x/x)/x)/z))=y.$x / ((((x/x)/y)/z) / (((x/x)/x)/z)) = y.$A morphism$(G,/) \to (G',/)$is a map$f : G \to G'$preserving$/$. This category is isomorphic to the category of groups, i.e. groups can be described by a single equation! If$G \in C$, then the corresponding group is$G$together with the multiplication$ab := a/((a/a)/b)$. If$G$is a group, then define$x/y = x y^{-1}$. Reference: Higman, Graham und Neumann, Bernhard: Groups as groupoids with one law, Publicationes Mathematicae Debrecen, 2 (1952), 215-227. Added 28/6/12: The category of A-spaces (topological spaces in which every intersection of open subsets is open) is isomorphic to the category of preorders. Under this isomorphism$T_0$A spaces correspond to partial orders. 4 edited body I just remember an example by myself. Let$C$be the following algebraic category: Objects are nonempty sets$G$together with a binary operation$/ : G \times G \to G$, such that for all$x,y,z \in G$, we have$x/((((x/x)/z)/(((x/x)/x)/z))=y.$x/((((x/x)/y)/(((x/x)/x)/z))=y.$

A morphism $(G,/) \to (G',/)$ is a map $f : G \to G'$ preserving $/$.

This category is isomorphic to the category of groups, i.e. groups can be described by a single equation! If $G \in C$, then the corresponding group is $G$ together with the multiplication $ab := a/((a/a)/b)$. If $G$ is a group, then define $x/y = x y^{-1}$.

Reference: Higman, Graham und Neumann, Bernhard: Groups as groupoids with one law, Publicationes Mathematicae Debrecen, 2 (1952), 215-227.

Added 28/6/12: The category of A-spaces (topological spaces in which every intersection of open subsets is open) is isomorphic to the category of preorders. Under this isomorphism $T_0$ A spaces correspond to partial orders.

3 added 228 characters in body

I just remember an example by myself.

Let $C$ be the following algebraic category: Objects are nonempty sets $G$ together with a binary operation $/ : G \times G \to G$, such that for all $x,y,z \in G$, we have

$x/((((x/x)/z)/(((x/x)/x)/z))=y.$

A morphism $(G,/) \to (G',/)$ is a map $f : G \to G'$ preserving $/$.

This category is isomorphic to the category of groups, i.e. groups can be described by a single equation! If $G \in C$, then the corresponding group is $G$ together with the multiplication $ab := a/((a/a)/b)$. If $G$ is a group, then define $x/y = x y^{-1}$.

Reference: Higman, Graham und Neumann, Bernhard: Groups as groupoids with one law, Publicationes Mathematicae Debrecen, 2 (1952), 215-227.

Added 28/6/12: The category of A-spaces (topological spaces in which every intersection of open subsets is open) is isomorphic to the category of preorders. Under this isomorphism $T_0$ A spaces correspond to partial orders.

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