That's purslane. It's edible (although I've never tried it), and seeds prolifically. I've had good luck pulling it, but you need to make sure to get the whole root: any small piece left behind will resprout. I don't use chemicals, but you can google to find out what might work in a lawn.

Thank you; I, too, have had some success with pulling the roots out but it seems like it's a losing battle. At best, I've kept them from taking over the entire lawn. I did google Purslane and you can actually by Purslane seeds. Maybe I should cultivate the weeds instead of trying to rid myself of them LOL

Haha - well it's not detrimental or hazardous like some weeds - no sticky sharp burs or anything.

It has multitudes of extremely fine seeds, so to eliminate, you need to get to it before it flowers - and expect old seeds to keep sprouting for some time.

Would a pre-emergent work on it, I wonder?

We had a great weed picking day, weather wise, last week. I spent a few hours pulling out all the Purslane that I could see. I reached down into the soil as deep as the weed root was. I'll keep monitoring the situation. Weeding should be much easier going forward. Thanks for all the advice.