[frud]

For basically forever compilers and languages have been designed with a total ordering on operator precedences. The Yacc compiler generator tool maps operator precedence to integers, as do most handwritten compiler parsers.

A total ordering has a definite answer to the question "is x greater than, equal to, or less than y?" for all x and y. A partial ordering will answer "I don't know" for some x and y.

It's quite possible to implement operator precedence using a partial ordering. When the parser has to resolve precedence between two operators and their relationship is not defined, throw an "ambiguous precedence" error.

You can implement the partial ordering by putting all the operators into a DAG of sets of operators. If two operators are in the same set, they have equal precedence. If there is a path through the DAG from one operator to the other, that defines the precedence relation. Else there is no relation.

Say "*" is defined to have a higher precedence than "+", and they both have an undefined precedence relation with "&". Then "1 + 2 * 3" should compile into "1 + (2*3)", but "1 + 2 & 3" should throw an "ambiguous precedence" syntax error.