The law of one price states that two portfolios that will produce exactly the same cash flows in the future must have the same value to start with. For example, a synthetic security should have the same price as the security it mimics. Similarly, the strips created from a gilt should be collectively worth the same as the original security.
This is a less demanding requirement than the non-existence of dominant trading strategies or no arbitrage. It is nonetheless useful in developing some parts of financial theory.
If there are no dominant trading strategies the law of one price holds but it is possible for dominant trading strategies to exist despite the law of one price holding.
As with arbitrage and dominant trading strategies we would expect the law of one one price to hold. If it did not hold, then market forces should bring prices back into line. No one would buy the more expensive portfolio if the cheaper one would produce the same cash flows, therefore the price of the constituents of the more expensive portfolio would come down.
However, there are occasional examples of failures of the law of one price and the existence of persistent arbitrage opportunities.