Hi, I cannot find an answer to this question, but it seems intuitively true. Say an operator is partial and associative in the following sense:
a * (b * c) = (a * b) * c if both sides are defined
That is, not the stronger constraint that "either both sides are defined and equal or both sides are undefined". In this case, is it true that all possible defined bracketings of a1 * a2 *.... * aN are equivalent? If so, what would be a proof argument? The standard induction-based proof used for total operators does not seem to work as some bracketings may not be defined:
Yes I asked if "all DEFINED bracketings" are equivalent, that is, there may be many bracketings of a1*....*aN of which some are defined and some are not. I want to know whether all the defined bracketings will be equal under this definition of associativity. Or if not, a counterexample will be helpful.