We introduce a notion of an associative trinary relation
arising from two homeomorphisms that satisfy a certain relation 
and construct an associated $C^*$-algebra.  
We prove a sufficient condition for the $C^*$-algebra to have a maximal 
closed two-sided ideal.  
As an example, we study a trinary relation 
arising from two homeomorphisms of the ring of $p$-adic integers 
and show that the associated $C^*$-algebra has a maximal closed two-sided ideal.