discriminator function
Let $A$ be a non-empty set. The ternary discriminator on $A$ is the ternary operation $t$ on $A$ such that
$$t(a,b,c):=\{\begin{array}{cc}a\hfill & \text{if}a\ne b,\hfill \\ c\hfill & \text{otherwise.}\hfill \end{array}$$ |
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
$$q(a,b,c,d):=\{\begin{array}{cc}d\hfill & \text{if}a\ne b,\hfill \\ c\hfill & \text{otherwise.}\hfill \end{array}$$ |
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:
$${t}_{1}(a,b,c):=\{\begin{array}{cc}b\hfill & \text{if}a\ne b,\hfill \\ c\hfill & \text{otherwise.}\hfill \end{array}\mathit{\hspace{1em}\hspace{1em}\hspace{0.5em}\hspace{1em}}{t}_{2}(a,b,c):=\{\begin{array}{cc}c\hfill & \text{if}a\ne b,\hfill \\ a\hfill & \text{otherwise.}\hfill \end{array}$$ |
But they are really no different from the ternary discriminator $t$:
$${t}_{1}(a,b,c)=t(b,a,c)\mathit{\hspace{1em}}\text{and}\mathit{\hspace{1em}}t(a,b,c)={t}_{1}(b,a,c),$$ |
$${t}_{2}(a,b,c)=t(c,t(a,b,c),a)\mathit{\hspace{1em}}\text{and}\mathit{\hspace{1em}}t(a,b,c)={t}_{2}(a,{t}_{2}(a,b,c),c).$$ |
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 |