Signatures

2020 James B. Wilson

Colorado State University

How is + different from -?

\(+\) takes two values, \(x+y\) whereas \(-\) really ought to be two different symbols \(x-y\) for example verses \(-y\).

To avoid confusion in algebra \(x-y\) is always short-hand for \(x+(-y)\) therefore the difference is \(+\) expects 2 values, \(-\) only one.

Languages have alphabets + grammars. Math grammars are often very basic, e.g.:

allow this \(x+y\) but not this \(+xy\).

In algebra grammar all we need is the number of terms to combine in an operator.

E.g. 2 in the case of \(+,\times\), 1 in the case of - and ^-1.

The signature of an operator is the number of terms it requires to be formed in the language grammar.

Signature of \(+, \times\) is 2 because the proper use involves 2 terms, \(x+y\) or \(x\times y\)

Signature of \(-\) and ^-1 is 1 because the proper use involves 1 terms, \(-x\) or \(x^{-1}\).

Suppose we a new operator define \([,]\) by \([x,y]=xy-yx\). What is its signature?

2, because it needs both an x and a y.

Suppose we have a new operator \(\langle,\rangle\) defined by \(\langle x,y,z\rangle=x(yz)-(xy)z\). What is its signature?

3, because it needs x, y, and z.

For an entire system of algebra the signature is the list of arities of all the operators involved.

E.g. \(\mathbb{N}\) under \(+,\times,0,1\) has signature \(2,2,0,0\).

What is the signature of \(\mathbb{M}_2(\mathbb{R})\) under \(+,-,0,\times,1\)?

2,1,0,2,0

Operators with signature 2 are called binary operators, they are the most common, e.g. +,\(\times,/\).
Operators with signature 1 are called unary, e.g. minus or inverse.
Operators with signature 0 are called nullary, these are thought of a constants as they do not require inputs to be formed but are fixed in the language. E.g. 0, 1 are nullary operators with often special roles.
Ternary and other "arity" signatures exist but are less frequently used.

"Infix" for a binary operation write \(x*y\)
"Prefix" for a binary operation write \(*xy\), for unary \(-x\) for example.
"Postfix" for a binary operation write \(xy*\), for unary \(x^{-1}\) for example.
"Outer notation" \([x,y]\) or \(\langle x_1,\ldots,x_{19}\rangle\). Used especially for large or unknown arity where compact infix notation makes no sense.

For different arities different conventions exist on how to write the operations for easy reading and detecting mistakes.