Answer: There are three tried-and-true candidates for the site you describe; Solerolia solerolii, Pratia angulata, and Sagina subulata.
Solerolia (Baby's Tears) grows 2-4" high and spreads by creeping stems, eventually formaing a lush, medium green mat. Sagina is called Irish or Scotch moss. These plants look like mosses, but they're really not, and prefer to grow in full sunshine rather than shade. Sagina takes some foot traffic and is available in either green or yellowish green foliage. Pratia (blue star creeper) is another prostrate plant, reaching only 2-3" in height. Foliage is dark green and the plants produce tiny, sky blue flowers in spring and summer.
Any of the above plants will grow nicely in your gardening region. Enjoy!
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