Constant-one operator

Read the equation

For an ordinary finite real exponent z, 1ᶻ equals one. The operation therefore erases its input value and replaces it with a constant. That loss of dependence is what makes the larger sequential operator composition collapse to one.

Terms & definitions

How it works

Read the definition from its right-hand expression to the quantity on the left. Its surrounding source records the intended input, state, or parameter context. Where the source leaves an operation unspecified, this page preserves that gap.

Assumptions & limits

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Original context

f1z=z+1,  f2z=z−1,  f3z=z.

f4z=z,  f5z=1,  f6z=z,  f7z=1/z.

Here f4z=z1, f5z=1z, and f6z=z/1. The reciprocal requires a nonzero input. An ordered composite can nevertheless be defined at an initial zero when a preceding operator supplies a nonzero reciprocal input.

6.1 Proposition 1: the sequential loop is a constant map

For every finite real initial value in the ordinary interpretation,

f7∘f6∘f5∘f4∘f3∘f2∘f1z=1.

{f}_{5}\left(z\right)={1}^{z}