Signature

An Algebra has a signature. The signature of an Algebra is a set of Function Symbols. We will designate a signature by with elements of this set represented by f.

Note: we need a fancy F for signature because the normal F is going to be a collection of operations.

Each f has an arity. An arity is a positive integer.

Appendix A

Examples are helpful.

• Groupoid, (G, *) has a signature of (2).
• Monoid (G, *, {1}) has a signature of (2,0).
• Group (G, *, , {1}) has a signature of (2,1,0).
  • It might not show up, that -1 is superscripted
• Semiring (R, +,*) has a signature of (2,2).
• Ring (R, +, *, -) has a signature of (2,2,1).
  • the symbol – with an arity of 1 is for commutative
• Field (K, +, *, {0}) has a signature of (2,2,1,0).

We found another example that we are currently exploring:

{one, zero, +, -}

Think about it, if you are closed to addition and have one in your set, then having n in your set means that n+1 is also in your set. If you are familiar with Induction you can see how having 1 gives you 2, having 2 gives you 3, etc, thus “grabbing” all the positive integers. Next, we have the option of subtracting a larger positive integer from a smaller positive integer, and that will take us into negative territory, thereby creating negative integers.

Appendix B

One comment seemed to say that the word ‘type’ might be used as a synonym for ‘signature’.

Appendix C

We found some more signatures and we like them for being about “stuff” that you’ve already learned.

• Z_add = (Z, +, -, 0)
• Z_mult = (Z, *, 1)
• Z_ring = (Z, +, *, -, 0, 1)

Appendix D

“A set of names of operations is called a signature.”