For "Off-Chain Orderbook, On-Chain Settlement" decentralized exchanges, final trades occur on chain but every order is hosted by a third party called a Relayer. This was the style of many early generation decentralized exchanges built on Ethereum.
- Faster order management than On-Chain solutions
- Less fees involved than On-Chain solutions
- No full guarantee that a maker order is still active
- The relayer could front run an order