An "F1" "Morkie" is a morkie that has a purebred Yorkie as one parent and a purebred Maltese as one parent. You can register these "designer puppies" with three different "designer dog" registries that do not try to say they are purebred dogs but that both parents are purebred dogs. These registries are United All Breed, American Canine Hybred Club, and American Canine association. There may be others. NO AKC will not recognize these dogs - but then again, there are purebred dogs that AKC took a long time in allowing in, such as Jack Russel's and Cavaliers.
The reason people want a purebred dog is that each purebred dog will have certain traits that you can be pretty certain will be in that purebred puppy if the purebred is bred correctly. When you get a Morkie - you are going to want to look at why a Morkie might be more the dog for you then a Purebred Yorkie or Maltese. Yorkies and Maltese are both very similar in a lot of ways: Size, grooming, etc. The very biggest difference that I have found in a Maltese vs a Yorkie - I have a Yorkie as a house pet and my daughter has a Maltese - and I have seen this same thing in other's Maltese and Yorkies - is that a Maltese (usually) really wants to please you - because a Maltese wants to please you, they tend to be more easily trainable - a Yorkie (usually) tends to want to be the one to be pleased - thus sometimes making a Yorkie much harder to train, especially to house break. I'd never give my little Yorkie girl up - she's my baby after all my "babies" have left home making me an "empty nester", but at 3 years old, she is "her own little dog" and will not house break. I'm sure there are some Yorkies that do house break easier and some Maltese that are harder to break.
The other thing about combining - and this might get some "slack" from those that are "purebred only" people, is that when you increase the "gene pool" you will have a much higher likely hood that you WILL NOT get the problems that just tend to be for Yorkies or just tend to be for Maltese. You can argue to the "sky" about this but the vets have a name for that and any vet will be able to explain to you why can actually will make a healthier dog by breeding two purebred dogs. They do this with other animals also to make a "stronger" animal with less health problems and other weaknesses. Almost all (not completely all) the bad traits of any dog will tend to be recessive (back dew claws are dominate) so if you have purebred dogs you will have a larger chance to get 2 recessive genes when breeding the purebred. You might not have realized you even had a parent dog with this recessive gene unless you sorry to say - breed with another dog that has this recessive gene. Then you will possibly have some puppies - it's based on the genes that the puppy gets - that will then have what ever the problems was from the two recessive genes coming together. Out of 4 puppies you might just have two puppies that have a problem and a puppy that will not have any recessive gene and a puppy that has only one recessive gene - you can get any number of different combinations like this in the 4 puppies. Genetics was my favorite part of college biology and I still tend to delve into it and I find it very interesting.
Morkies are very cute and they can be more like one of the two breeds in looks while being more like the other breed in personality. The personality part is not one that you will know about as easily when you just get a 10 to 12 week old puppy. I do know people that have Morkies and they are as happy with babies as the people I know that have and love their Yorkie babies.
Hope this helps.