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Lincoln never held a seat in the US Senate, but he ran for it in 1858 against Douglas.
In politics, the incumbent is the person who holds the seat he/she is running for in order to keep it. The challenger is the person trying to win the seat that the incumbent has.
Single Member voting is simply what we do when we cast our vote for a particular candidate. The winner is the one with the most votes. There is no need to make a certain percentage in order to win At Large Voting is when there would be more than one seat voted on: for example 5 open seats in an election. Usually along the lines of a school board or other types of municipal elections.
The President can never appoint someone to fill a Senate vacancy. The U.S. Constitution allows the Governor of the state to which the vacant seat is entitled to make a temporary appointment to fill the seat until a special election can be held, but only if the state legislature has passed a bill allowing him/her to do so. Actually there is one Senate vacancy for which the President appoints a replacement: the U.S. Vice President, who constitutionally is also the President of the Senate. Vice presidential appointments require the approval of both houses of Congress.
It is fair that people of a congressional district might elect a candidate then have the House of Representatives vote not to seat that person.