Race condition
Race conditions arise in software when an application depends on the sequence or timing of processes or threads for it to operate properly. As with electronics, there are critical race conditions that result in invalid execution and bugs. Critical race conditions often happen when the processes or threads depend on some shared state. Operations upon shared states are critical sections that must be mutually exclusive. Failure to obey this rule opens up the possibility of corrupting the shared state.
Ore condition
The Ore condition can be generalized to other multiplicative subsets, and is presented in textbook form in and . A subset "S" of a ring "R" is called a right denominator set if it satisfies the following three conditions for every "a", "b" in "R", and "s", "t" in "S":
If "S" is a right denominator set, then one can construct the ring of right fractions "RS" similarly to the commutative case. If "S" is taken to be the set of regular elements (those elements "a" in "R" such that if "b" in "R" is nonzero, then "ab" and "ba" are nonzero), then the right Ore condition is simply the requirement that "S" be a right denominator set.