I am looking for examples of varieties of Universal Algebras that admit a ternary majority term $p$. For example, boolean-algebras have such a term:
- $p(x,y,z) = (x \wedge y)\vee (x \wedge z) \vee (y \wedge z)$
After looking through standard references, I have only found a few examples of such varieties, most of them are arithmetical varieties. If you have any examples, they would be much appreciated.
As Pedro S. Terraf points out in his comment, the term you present is valid for lattices, not just for Boolean algebras.
Likewise, it works for every variety with a lattice reduct.
Some of these varieties are not arithmetical. Example: the variety of ortho-lattices (but the one of ortho-modular lattices is arithmetical). One example of a not congruence-permutable ortho-lattice is the eight-element of length $4$.
If you want some which are not lattice-based, consider the following example, which provides in the most trivial way a ternary majority term:
$\mathbf{A}$ is an algebra with a single ternary operation $g$ which satisfies $g(x,x,y) \approx x$; if $x \neq y$, let $g(x,y,z) \approx z$ (just to give a complete description of the operation). Then clearly $\mathbf{A}$ admits a ternary majority term — the fundamental operation $g$ — and so the same happens with the variety that $\mathbf{A}$ generates. I don't know whether or not this variety is arithmetical.
For another "exotic" example consider the Example 1.4 here: http://spot.colorado.edu/~kearnes/Papers/idem.pdf
There, the author claims that the variety is congruence-distributive (although that doesn't necessarily entail the existence of a majority term, so I don't know if there is one). Again, I also don't know if the variety is arithmetical.