multisorted algebra | Gyre&Gimble

Operation: Is it just a name or is there a metaphor behind it?

A function of the form ${f:S\times S\rightarrow S}$ may be called a binary operation on ${S}$ . The main point to notice is that it takes pairs of elements of ${S}$ to the same set ${S}$ .

A binary operation is a special case of n-ary operation for any natural number ${n}$ , which is a function of the form ${f:S^n\rightarrow S}$ . A ${1}$ -ary (unary) operation on ${S}$ is a function from a set to itself (such as the map that takes an element of a group to its inverse), and a ${0}$ -ary (nullary) operation on ${S}$ is a constant.

It is useful at times to consider multisorted algebra, where a binary operation can be a function ${f:S_1\times S_2\rightarrow S_3}$ where the ${S_i}$ are possibly different sets. Then a unary operation is simply a function.

Calling a function a multisorted unary operation suggest a different way of thinking about it, but as far as I can tell the difference is only that the author is thinking of algebraic operations as examples. This does not seem to be a different metaphor the way “function as map” and “function as transformation” are different metaphors. Am I missing something?

In the 1960’s some mathematicians (not algebraists) were taken aback by the idea that addition of real numbers (for example) is a function. I observed this personally. I don’t think any mathematician would react this way today.

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Operation as metaphor in math

math, language and other things that may show up in the wabe