User:Dfeuer/Definition:Cone Compatible with Operation

Definition

Let $\struct {S, \circ}$ be a magma.

Let $C$ be a subset of $S$.

Suppose that for each $x,y \in S$:

$(1): \quad$ If $x, y \in C$ then $x \circ y \in C$

$(2): \quad$ If $x \circ y \in C$ then $y \circ x \in C$.

Then $C$ is a cone compatible with $\circ$.

Note that the second condition is trivially satisfied when $\circ$ is commutative, as when it is the operation of an abelian group or the addition operation of a ring.