Scalar extension

Scalar extension is a way to apply a function with a scalar argument when an array of a particular non-empty shape would be expected. The scalar is extended to this shape by treating it as an array with each element equal to the scalar's only element. This is equivalent to reshaping the scalar to fit the desired shape.

History and terminology
The concept of scalar extension has been around since APL\360. An example which extends the scalar  is: A Programming Language describes the above computation as a "scalar multiple" but does not generalise it to arbitrary scalar functions, so it's unclear when scalar extension as a unified concept was adopted in Iverson notation.

The word "extension" applies to scalar extension in two ways: first, a function is extended by making a case which would have been a RANK ERROR into a valid application. Second, the application works by conceptually extending the scalar to function as though it were an array of higher rank.

Two arrays are said to conform if they have the same shape or at least one can be extended (it is a scalar, or, in langauges with singleton extension, has exactly one element). A pair of conforming arrays defines a single shape which describes how their elements are paired: if neither is a scalar, it is their shared shape; if one is a scalar, it is the other's shape; if both are scalars, it is the empty vector,  (Zilde).

Rank extension
The term "scalar extension" is sometimes used to refer to the practice of allowing a scalar when a higher rank is expected. The scalar is treated as an array of the expected minimum rank whose shape is a vector of 1s (that is, a singleton). For example,  and   both produce an 8-element array even though the shape of an array is always a vector and so cannot be equal to. This type of extension, which differs from ordinary scalar extension in that there is no expected shape and only an expected rank, has also been present since APL\360.

Singleton extension
Some APLs, such as Dyalog APL, treat arrays with one element (singletons) as scalars for the purposes of scalar extension. This practice is referred to as "singleton extension". For example,

In this case addition accepts a singleton, and discards its shape. If two singletons are used as arguments, they are still considered to conform; the shape of the result is taken from the argument with higher rank.

Singleton extension can sometimes conflict with other extensions, an issue which does not occur with scalar extension. If a function is extended to allow a shorter vector argument to be extended (perhaps by padding with 0), but it also supports singleton extension, then there is a conflict with length-1 vectors.

Extension in the Rank operator
The Rank operator uses a generalization of scalar extension to pair cells. A function called with rank 0 exhibits ordinary scalar extension: it acts like a scalar function. A function with higher rank extends not scalars (arrays with empty shape) but arrays whose frame is empty. An empty frame implies there is only one cell, and it has a scalar-like array structure. This cell can be extended by reusing it for every function call.

Examples
Dyadic scalar functions and the Each operator use scalar extension to pair their arguments:

Replicate and Partitioned Enclose (and also Partition, but this is less useful, as it is equivalent to  or   ) extend a scalar left argument to apply to each column of the right argument:

Decode uses extends a single radix left argument to apply to all digit values in the right argument:

APL2 and Dyalog APL use a variant of singleton extension when the selected axis of the right argument has length one: each element along that axis is reused for every element of the left argument.

In languages which allow a vector left argument to Rotate, the behavior with a scalar left argument follows from scalar extension. In the following example a length-2 vector could be used to rotate each row by a different amount. A scalar rotates both rows by the same amount.

Each left and each right
A common problem is pairing up each element of one argument with all the elements of another argument. Because of scalar extension, making one argument a scalar accomplishes this: For scalar functions, explicit use of the Each operator is unnecessary: Binding one argument to the function also works, but this always requires the Each operator: