It is possible to reinterpret equivalence relations and partial orders as operations of some arity?

Question

I have a question going on in my head for some days. To better write it, I need to know if the axiomatizations of partial orders, equivalence relations and so on may be rewritten in terms of I-ary operations (I may be also an infinite index set). I'm not an expert on universal algebra, so I ask for a detailed answer and for references, if any.

Answer 1

Given a reflexive relation $R$ on a set $S$, you can define a binary operation $*$ on $S$ by $a*b=a$ if $aRb$ and $a*b=b$ otherwise, and conversely you can recover $R$ from $*$ (given that $R$ is known to be reflexive). So, you could rewrite all the axioms for a partial order or an equivalence relation in terms of the corresponding binary operation $*$.

(If you allow operations that are not total, you can drop the requirement that $R$ is reflexive by saying $a*b=a$ if $aRb$ and otherwise $a*b$ is undefined. With total operations, though, there is no way to encode arbitrary relations on a set in general, for the simple reason that a singleton set has only one total operation of any arity but has two different relations.)