discriminator function

Let $A$ be a non-empty set. The ternary discriminator on $A$ is the ternary operation $t$ on $A$ such that

t(a,b,c):=\left\{\begin{array}[]{ll}a&\textrm{if }a\neq b,\\ c&\textrm{otherwise.}\end{array}\right.

In other words, $t$ is a function that determines whether or not a pair of elements in $A$ are the same, hence the name discriminator.

It is easy to see that, by setting two of the three variables the same, $t$ becomes a constant function: $t(a,b,a)=a$ , $t(a,a,b)=b$ , and $t(a,b,b)=a$ .

More generally, the quaternary discriminator or the switching function on $A$ is the quaternary operation $q$ on $A$ such that

q(a,b,c,d):=\left\{\begin{array}[]{ll}d&\textrm{if }a\neq b,\\ c&\textrm{otherwise.}\end{array}\right.

However, this generalization is really an equivalent concept in the sense that one can derive one type of discriminator from another: given $q$ above, set $t(a,b,c)=q(a,b,c,a)$ . Conversely, given $t$ above, set $q(a,b,c,d)=t(t(a,b,c),t(a,b,d),d)$ .

Remark. The following ternary functions $t_{1},t_{2}:A^{3}\to A$ could also serve as discriminator functions:

t_{1}(a,b,c):=\left\{\begin{array}[]{ll}b&\textrm{if }a\neq b,\\ c&\textrm{otherwise.}\end{array}\right.\hskip 28.452756ptt_{2}(a,b,c):=\left\{% \begin{array}[]{ll}c&\textrm{if }a\neq b,\\ a&\textrm{otherwise.}\end{array}\right.

But they are really no different from the ternary discriminator $t$ :

t_{1}(a,b,c)=t(b,a,c)\quad\mbox{ and }\quad t(a,b,c)=t_{1}(b,a,c),

t_{2}(a,b,c)=t(c,t(a,b,c),a)\quad\mbox{ and }\quad t(a,b,c)=t_{2}(a,t_{2}(a,b,% c),c).

References

1 G. Grätzer: Universal Algebra, 2nd Edition, Springer, New York (1978).
2 S. Burris, H.P. Sankappanavar: A Course in Universal Algebra, Springer, New York (1981).

Title	discriminator function
Canonical name	DiscriminatorFunction
Date of creation	2013-03-22 18:20:58
Last modified on	2013-03-22 18:20:58
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	6
Author	CWoo (3771)
Entry type	Definition
Classification	msc 08A40
Synonym	switching function
Defines	ternary discriminator
Defines	quaternary discriminator