You do not necessarily need a mat under a tent. If your tent has its own 'floor', sometimes having a mat under it is a good idea to prevent it from being torn by sharp objects on the bare ground or inside the tent. If it has no floor, and your tent is on sand or gravel, a mat is nice to have because it makes the floor cleaner and more comfortable for bare feet, and may reduce the chance of insect bites.
under a mat
The mat that is placed under a Bunsen burner can be referred to as a heat proof mat.
Waterproof layer spread under a sleeping bag in a tent?
gause mat
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A lean-to is a building with three sides and a sloping roof. Its use is not primarily to house a tent, but I suppose if it was big enough, then you could set one up under it. Lean-to's are generally small and use for storage or shelter. A tent is shelter, itself; there's no need to build a lean-to around it unless you want to keep the dew/rain off the tent.
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It slides under the first mat section
It depends on how tall the tent is.
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under the door mat
present ham peasant Christ cat can sat mat sent set tent pet mast hat