I have come accross something called operator precedence grammar https://en.wikipedia.org/wiki/Operator-precedence_grammar and I would like to know about the specific mathematical properties is presents. In particular, I would be interested in knowing whether it is a kind of Polish notation (if not, in which ways it deviates from that), and also whether it might be useful for representing right ideals in a ring.
Thanks in advance.
From what I have gleaned here, an operator precedence grammar is a preorder on a grammar with special properties (outlined in the first link). The idea is that they define something like the order of operations for arithmetic expressions in integers (where multiplication takes precedence over addition, etc.) More broadly speaking it is a special binary relation on the grammar.
The grammar is not a notation any more than the statement "$2$ is less than $4$" is a notation. If you are asking whether or not authors use any sort of Polish notation when they talk about precedences, then the answer is maybe. An author can choose to use such a notation or not. The book I linked to above uses infix notation, not prefix or postfix notation. In any case, notation does not change the meaning of what an operator precedence grammar is.
I am a ring theorist, but I haven't heard of any connections to this, nor can I immediately spot any connection to ideals or rings.