-ary set operations.

1. Jul 7, 2009

matheinste

Hello all.

Along the lines of unary and binary operations, could someone describe what 0-ary operation is and possibly give an example. I have only seen such an operation mentioned once, and that was early in the first chapter of Grillet's Abstract Algebra (2007), and so presumably it does not figure figure very highly, but is of some interest. The explanation in the above mentioned book was not very clear to me.

Thanks. Matheinste.

2. Jul 7, 2009

g_edgar

A 0-ary (or "nullary") operation is also known as a "constant symbol".

Example. The first-order logical theory of "fields" can be formulated in many ways.One of them has...
two binary operations:
$$(x,y) \mapsto x+y$$
$$(x,y) \mapsto x\cdot y$$
two unary operations:
$$x \mapsto -x$$
$$x \mapsto 1/x$$
and two nullary operations:
$$0$$
$$1$$
then some axioms that these operations must satisfy. Actually, $$1/x$$ is not defined for $$x=0$$, so some adjustment would have to be made for that.

3. Jul 7, 2009

matheinste

In the case you have cited are the zero and one nullary operators the same as the identity elements for the two field operations?

Matheinste.

4. Jul 7, 2009

g_edgar

Yes. Among the axioms would be
$$(\forall x)\; (x+0=x)$$
$$(\forall x)\; (x\cdot 1 = x)$$

5. Jul 7, 2009

matheinste

But aren't these binary operations.

Matheinste.

6. Jul 7, 2009

Moo Of Doom

+ and * are, but 0 and 1 aren't.

7. Jul 7, 2009

matheinste

Yes, so they are.

My problem, mental block, is that a binary operation on a set is an operation needing two inputs from the set which returns a member of the set. A unary operation has a single input and returns a member of the set. And so a nullary operation has no inputs and returns a member of the set. Does this equate to just picking a member of the set.

Matheinste.

8. Jul 7, 2009

CRGreathouse

A unary operation can be identified with its output, yes. That's why g_edgar wrote 0 and 1 instead of 0() and 1().

9. Jul 8, 2009

matheinste

Hello again

----A 0-ary or constant operation on a set S is a mapping f : {0} −→ S and simply selects one element f (0) of S. -----

Can anyone please explain what exactly does the "element f(0) of S" mean in the above sentence, from Grillet, Abstract Algebra, .

Matheinste.

10. Jul 9, 2009

honestrosewater

f(0) denotes the value assigned to 0 by f. The value will be a member of f's range, which is S.

11. Jul 9, 2009

matheinste

Does this leave us free to assign any member of the range, which will be a member of the set S, as the image of the function.

Matheinste.

12. Jul 11, 2009

honestrosewater

When you define the function, you can choose any value from S. But the requirement that a function assign only one value to each member of the domain means that you cannot assign any more values after one has been chosen. You would have to define a new function and give it a new name.