Example of social choice rule that does not satisly the unrestricted domain condition Can any social choice rule that is not complete said to be violating the unrestricted domain condition? Could you provide an example of SCR other than Pareto dominance that is not complete or violates unrestricted domain property?
Can any social choice rule that is not complete said to be violating the unrestricted domain condition?


Yes.


  Could you provide an example of SCR other than Pareto dominance that is not complete or violates unrestricted domain property? 


Let society consist of two individuals $i$ and $j$. Let there be exactly two alternatives $A$ and $B$.

Each individual has 3 possible preferences: strictly prefer $A$ to $B$, indifferent between $A$ and $B$, or strictly prefer $B$ to $A$.

Hence, there are 3 × 3 = 9 possible social preference profiles:


$i$ strictly prefers $A$ to $B$ and $j$ strictly prefers $A$ to $B$.
$i$ strictly prefers $A$ to $B$ and $j$ is indifferent between $A$ and $B$.
$i$ strictly prefers $A$ to $B$ and $j$ strictly prefers $B$ to $A$.
$i$ is indifferent between $A$ to $B$ and $j$ strictly prefers $A$ to $B$.
$i$ is indifferent between $A$ to $B$ and $j$ is indifferent between $A$ and $B$.
$i$ is indifferent between $A$ to $B$ and $j$ strictly prefers $B$ to $A$.
$i$ strictly prefers $B$ to $A$ and $j$ strictly prefers $A$ to $B$.
$i$ strictly prefers $B$ to $A$ and $j$ is indifferent between $A$ and $B$.
$i$ strictly prefers $B$ to $A$ and $j$ strictly prefers $B$ to $A$.


We say that a social choice rule (SCR) is complete or satisfies the unrestricted domain condition if it maps (or assigns) each of the above 9 profiles to a preference. (A trivial example of such a SCR is that which maps each of the above 9 profiles to the preference "strictly prefer $B$ to $A$".)

A trivial example of a SCR that is incomplete or violates the unrestricted domain condition is that which maps each of profiles 1–8 to the preference "strictly prefer $B$ to $A$", but fails to map profile 9 to any preference. (Another trivial example is @Giskard's: The SCR which fails to map any of the 9 profiles to any preference.)