What are the implications of "Law of one price" ? Is it related to Purchasing Power Parity ?

Yes, these both are of similar terms. Law of one price goes for same prices in every country of the same product that is being shifted by demand for low price product. when the demand of a second product that has less price will increase then the price will also move up. but price for the first decreases due to decrease in demand. Hence the prices of both will come into equilibrium that is known to be Law of One Price.

the real purchasing power of different currencies tends to equalize inflation rate differentials between their respective countries, resulting in nearly the same price for commodities (such as oil) in every country.

Law of one price is related to the foregin currecy transactions. It means that prices of same commodity will be same in all countries. So if the markets are efficient, there is no trnasportation costs, no transaction costs, then the difference of prices is only because of the inflation rate difference.

 

So a commodity will be prices differently in a country having more inflation than in the country where the inflation rate is less. This concept can be generalized even to the currecy prices. So the currency prices are also based on the inflation rates. So if we know what will be the inflation rate in a country we can estimate the currency prices, in future. The formulae is

 

The future currency rate currency A/B = current rate currency A/B X (1+inflation rate country of currency B) divided by (1+ inflation rate country of currency A)