door-mat (n.)

"movable barrier, commonly on hinges, for closing a passage into a building, room, or other enclosure," c. 1200, a Middle English merger of two Old English words, both with the general sense of "door, gate": dor (neuter; plural doru) "large door, gate," and duru (fem., plural dura) "door, gate, wicket." The difference (no longer felt in Old English) was that the former came from a singular form, the latter from a plural.

Both are from Proto-Germanic *dur-, plural *dures (source also of Old Saxon duru, Old Norse dyrr, Danish dr, Old Frisian dure, dore, dure, Old High German turi, German Tür). This is from PIE root *dhwer- "door, doorway."

Middle English had both dure and dor; the form dore predominated by 16c. but was supplanted later by door. The oldest forms of the word in IE languages frequently are dual or plural, leading to speculation that houses of the original Indo-Europeans had doors with two swinging halves.

Figurative sense of "means of opportunity or facility for" was in Old English. Phrase door to door "house to house" is from c. 1300; as an adjective, in reference to sales, by 1902.

A door is what a dog is perpetually on the wrong side of. [Ogden Nash]

"an article plaited or woven of more or less coarse natural materials (rushes, straw, twine, etc.) used as bedding, floor-coverings, etc.," Old English matte, from Late Latin matta "mat made of rushes" (4c.), probably from Punic or Phoenician matta (compare Hebrew mittah "bed, couch").

Meaning "tangled mass; anything closely set, dense, and thick" is from 1835. Meaning "thin, flat article to be placed under a dish, plate, etc. to protect the table" is by 1800. That of "piece of padded flooring used in gymnastics or wrestling" is attested from 1892; hence figurative phrase go to the mat "do battle" (1910). The Latin word also is the source of German Matte, matze; Dutch mat, Italian matta. French natte "mat, matting" is from Late Latin secondary form natta (compare napkin).