It is strongly recommended (though not required) that natural orderings be consistent with equals. This is so because sorted sets (and sorted maps) without explicit comparators behave "strangely" when they are used with elements (or keys) whose natural ordering is inconsistent with equals. In particular, such a sorted set (or sorted map) violates the
general contract for set (or map), which is defined in terms of the equals operation.
For example, if one adds two keys a and b such that (a.equals((Object)b) && a.compareTo((Object)b) != 0) to a sorted set that does not use an explicit comparator, the second add operation returns false (and the size
of the sorted set does not increase) because a and b are equivalent from the sorted set's perspective.
Virtually all Java core classes that implement comparable have natural orderings that are consistent with equals. One exception is java.math.BigDecimal, whose natural ordering equates BigDecimals with equal values and different precisions (such as 4.0 and 4.00).