Read the equation
The first operator adds one, the second subtracts one, and the third leaves its input unchanged. Applying the first two in sequence restores any ordinary finite real input. Their names do not alter these elementary operations.
Terms & definitions
| Term | Meaning in this source |
|---|---|
| f_1 | Increment by one |
| f_2 | Decrement by one |
| f_3 | Identity |
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
Read the surrounding source passage
This section audits the ordinary scalar interpretation, not every possible stateful implementation that could later be designed. Changing scalar operators into memoryful, vector-valued systems requires additional definitions. We do not silently add those definitions and then attribute their behavior to the original seven expressions. Let 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
Exact original notation
{f}_{1}\left(z\right)=z+1, {f}_{2}\left(z\right)=z-1, {f}_{3}\left(z\right)=z.Sources & provenance
{f}_{1}\left(z\right)=z+1, {f}_{2}\left(z\right)=z-1, {f}_{3}\left(z\right)=z.
{f}_{1}\left(z\right)=z+1, {f}_{2}\left(z\right)=z-1, {f}_{3}\left(z\right)=z.