If you only consider one semigroup, monoid or group, or if you consider only a particular one and its subsemigroups, submonoids or subgroups, then it’s convenient to use infix notation, with either \({ · }\) or an empty string rather than functional notation. This makes some of the equations above look more familiar. \[ f ( x ^ m , x ^ n ) = x ^ { m + n } , \] for example, becomes \[ x ^ m · x ^ n = x ^ { m + n } \] or just \[ x ^ m x ^ n = x ^ { m + n } . \] Also, because of the generalised associativity property, we don’t need parentheses to indicate the order of operations, so we can write expressions like \[ x y x ^ { - 1 } y ^ { - 1 } \] without specifying which of the five possible orders of operations are intended. When using this notation there are two different conventions for the identity element. Some authors use \({ 1 }\) and some use \({ e }\).
This notation is less cumbersome than functional notation, and much less cumbersome than the relational notation from the set theory chapter, but it can be confusing in two situations. One is where we have multiple semigroups, each with its own binary operation. The other is where symbols like \({ · }\) or \({ 1 }\) have previously established meanings which conflict with the usage here, as when discussing the integers with their additive structure.