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Revision difference : idempotency |
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| If $(S,*)$ is a magma, then an element $x\in S$ is said to be \emph{idempotent} if $x*x=x$. If every element of $S$ is idempotent, then the binary operation $*$ (or the magma itself) is said to be idempotent. For example, the $\land$ and $\lor$ operations in a lattice are idempotent, because $x\land x = x$ and $x\lor x = x$ for all $x$ in the lattice. |
If $(S,*)$ is a magma, then an element $x\in S$ is said to be \emph{idempotent} if $x*x=x$. If every element of $S$ is idempotent, then the binary operation $*$ (or the magma itself) is said to be idempotent. For example, the $\land$ and $\lor$ operations in a lattice are idempotent, because $x\land x = x$ and $x\lor x = x$ for all $x$ in the lattice. |
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A function $f\colon D\to D$ is idempotent if $f\circ f=f$. (This is just a special case of the above definition, the magma in question being the monoid of all functions from $D$ to $D$, with the operation of function composition.)
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A function $f\colon D\to D$ is idempotent if $f\circ f=f$. (This is just a special case of the above, definition, the magma in question being the monoid of all functions from $D$ to $D$, with the operation of function composition.)
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| In other words, $f$ is idempotent iff repeated application of $f$ has the same effect as a single application: $f(f(x)) = f(x)$ for all $x\in D$. |
In other words, $f$ is idempotent iff repeated application of $f$ has the same effect as a single application: $f(f(x)) = f(x)$ for all $x\in D$. |
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