OK, I give up. What's "lamination"?
Basically, slate is hard enough that it's not wearing down from foot traffic. I agree with the previous post that suggested that there was some sort of finish put on it that gave it a glossy appearance that is now wearing down.
If that's the case, then it can be maintained just the same way as a VCT tile floor, where you periodically scrub off the dirt-embedded surface layer of finish, and apply a new coat of finish.
You CAN do this yourself, but it would require a substantial investment of $300 to $1200 for a used or new floor machine and $200 for a rotary brush.
This is a typical floor machine: (and it's similar to my own)
And, for that floor machine you can buy all sorts of attachments to do everything from sanding down wood gymnasium floors to cleaning carpets with a wool "bonnet" to scrubbing rough floors (like ceramic tile or slate) to stripping finish off of granite floors. The way you would clean a ceramic tile or slate floor would be with a rotary brush like the one shown below:
And, of course, rotary brushes for floor machines come in a myriad of different bristles, from very soft (for polishing floors) to stiff abrasive impregnated bristles for very aggressive scrubbing. It's those millions of bristles that get into every little crack and crevice on a rough floor or ceramic tile floor.
The person to talk to regarding the kind of finish to use on your slate floor would be the local S. C. Johnson Wax sales rep. He can be found by phoning around to the places listed under "Janitorial Equipment & Supplies" in your yellow pages phone book. Ask each place if they sell "Johnson Wax" floor finishes. If so, ask them for the name and phone number of the Diversey sales rep.
You see, the S. C. Johnson Wax Company still makes floor "waxes" for residential use, but they split off their professional division into a separate company called "Johnson Professional". Johnson Professional then merged with the floor maintenance division of the Diversey Lever Company to form a company called "Johnson Diversey". Just this last spring, Johnson Diversey was purchased by Diversey Lever and is being operated under the name "Diversey". So, if you get the name and phone number of the local Diversey rep, then he used to work as a sales rep for the S. C. Johnson Wax Company. The S. C. Johnson Wax Company is well respected in the janitorial sector of the economy because they make quite good products, but they also train their sales reps well. (They have to. Some of the floors their products get put down on are so big (think airport, shopping center, convention center) that if the wrong stuff gets put down, it might take a judge to decide who pays the cost of taking the wrong stuff off, and putting the right stuff on.)
So, if you want to save money, the way to proceed is as follows:
1. Phone around to the places listed under "Janitorial Contractors" in the yellow pages, and ask if they have a floor machine. 90% of them will have several.
2. If they have a floor machine, ask if they have a rotary scrub brush suitable for cleaning a slate floor. If so, then they can do all the work without you're having to buy anything. If not, then you might have to buy a rotary scrub brush for about $200 and let them use it.
3. Let them clean the floor with the brush, recover the soiled cleaner with a wet/dry vaccuum, spray or mop down clean rinse water and then recover the soiled rinse water with a wet/dry vaccuum. Allow time for the floor to dry, and then you mop on multiple coats of the floor finish the Diversey rep recommended. You CAN use an ordinary mop to do this, but be sure to wrap the mop head tightly in a plastic bag between coats to prevent the finish from drying out in the mop head. Janitorial supply stores will also sell special finish applicators that you can use that basically consist of a dust mop with a sponge inside it for applying liquid floor finishes of various sorts. Alternatively, you could also use a "lamb's wool" which are normally used for varnishing hardwood floors. All would work well for you.
The two tricks are:
1. Soak the applicator in water and squeeze it out before using it to apply any water based finishes. This will prevent the water based finish from foaming up on the applicator.
2. Wrap the applicator tightly in a plastic bag (clear with no printing on it, or remove the printing with nail polish remover) to prevent it from drying out between coats. The coalescing solvents in the floor finish will dissolve some printing inks, with the result that the dyes from the printing on the bag will end up in your floor finish, giving you a slightly coloured floor finish. You can do without that.
3. Apply LOTS of finish. The thicker the coat of protection on the floor, the better the floor is protected from wear & tear, scratches and stains. You're better off with the finish on the floor than inside the jug.
I recommend that YOU apply the floor finish. That's because if you ask for quotes from janitorial companies to apply the finish, they will give you a price that includes TWO finish coats, which simply isn't enough to protect the floor. The reason why their quote will be for only two finish coats is because the costs escalate rapidly for each additional coat. The reason why is that the janitors can mop one finish coat on immediately after vaccuuming up the soiled rinse water. Then while that first coat is drying, they can be cleaning their wet/dry vaccuum and putting their equipment back in the truck. Then, they can apply another finish coat immediately before they leave.
If you want a third and fourth finish coat, then you're paying the janitors to sit around and play cards while each coat is drying. So, to get the job, the janitorial company is going to trim off as much idle sitting around as possible, and quote the job presuming they apply two, and only two, finish coats. That's not enough to protect the floor or even give you a nice gloss on the floor, so mop it on yourself, and don't stop mopping it on until you see a real nice gloss on your floor.