Can any mathematical relation be called an 'operator'? Mathematics authors agree that $+,-,/,\times$ are basic operators. There are also logical operators like $\text{or, and, xor}$ and the unary negation operator $\neg$.
Where there seems to be a disagreement, however, is whether certain symbols used in the composition of propositions qualify as operators. By this definition:

an operation is an action or procedure which produces a new value from
  one or more input values, called "operands"

symbols that denote a relation should also be operators. 
For instance, $a<b$ takes integral operands and returns a value in the Boolean domain. Even something like $x \in S$ should be an operator that takes an element and a set, and also returns a truth value. The same could be said for the equality operator $=$.
Computer scientists call these 'relational operators', but mathematicians rarely do this. Is there a reason for this discrimination?

A serious disadvantage of treating logical connectives as operators returning a value (that is, as functions) is that in non-classical logics their "range" may fail to correspond to any two-element set $\{\top,\bot\}$, and according to the Wikipedia page Truth value, may even fail to correspond to any set whatsoever: "[n]ot all logical systems are truth-valuational in the sense that logical connectives may be interpreted as truth functions. For example, intuitionistic logic lacks a complete set of truth values...." (Note that in particular, the naive attempt to consider logical connectives as mapping into a three-element set $\{\top,\bot,\text{some third value}\}$ does not work.)
