This link gives you the basic idea. I think there are several different products on the market that are similar, but vary price (& quality) wise.

https://www.familyhandyman.com/decks/building-a-deck/under-deck-roof/view-all/

That makes no sense if they want the patio to stay dry. I think Duradeck could work but you will need to contact them

The OP is asking for the patio "underneath" to stay dry, and for the cedar decking (above) to look decent. Maybe the question isn't completely clear, but I didn't get the impression that they wanted to cover up the cedar.

Maybe they will clarify...

Instead of building a permanent structure underneath, I was wondering if I can put something on top just for the wet winter, and take it out in summer, e.g., the orange waterproof tile membrane, vinyl floor sheets?

Anything I can use that fits the idea and looks decent?

We don't use the deck in winter, and there is almost no rain in summer. The deck looks beautiful. Here is a picture of it being stained.

Thanks for all your advices.

JoAnn

There are several vendors showing their product (like Mary Elizabeth suggested) at the Washington State Fair in Puyallup until 24 Sept.

Indoor outdoor carpet with rubber marine backing?

Keeping the patio dry (barely sloped roof over patio but under deck) is a separate issue from keeping the deck looking good (stain).