Addition and multiplication do have the identities and , respectively. All varieties of addition and multiplication are commutative over the floats, but the lack of associativity causes any non-trivial symbolic manipulation of an expression to affect the expression's value.
Negation is a total inverse for all three addition operators.
Since the base is a fixed integer none of the three multiplication operators have total inverses. None of the possible distributive laws are obeyed. For many applications the use of floating point does not adversely affect the output, which has been envisioned as coming from computations using real numbers. With long streams of computations there is a worry that the floating point computation stream will radically diverge from the underlying real computation stream. In these cases, formal arguments involving particular implementations of the operators and particular sequences of computations must be made.