Etymology
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mate (n.1)

mid-14c., "associate, fellow, comrade;" late 14c.,"habitual companion, friend;" from Middle Low German mate, gemate "one eating at the same table, messmate," from Proto-Germanic *ga-matjon, meaning "(one) having food (*matiz) together (*ga-)." For *matiz, see meat. It is built on the same notion as companion (which is thought to be a loan-translation from Germanic). Cognate with German Maat "mate," Dutch maat "partner, colleague, friend."

Meaning "one of a wedded pair" is attested from 1540s. Used as a form of address by sailors, laborers, etc., at least since mid-15c. Meaning "officer on a merchant vessel" is from late 15c.; his duty is to oversee the execution of the orders of the master or commander.

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mate (v.1)

c. 1500, transitive, "to equal, rival," 1590s as "to match as mates, couple, join in marriage," from mate (n.1). Also, of animals, "to pair for the purpose of breeding" (c. 1600). Intransitive sense of "be joined in companionship" is from 1580s. Related: Mated; mating.

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mate (v.2)

"to checkmate," c. 1300, from Old French mater "to checkmate, defeat, overcome," from mat "checkmated" (see checkmate (v.)).

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mate (n.2)

in chess, "a condition of checkmate, the state of the king when he is in check and cannot move out of it," c. 1300, mat, from Old French mat, from mater "to checkmate" (see mate (v.2)).

Fool's mate, a mode of checkmate in which the tyro, moving first, is mated by his opponent's second move.—Scholar's mate, a simple mode of checkmate, sometimes practised on inexperienced players, in which the skilled player's queen, supported by a bishop, mates the tyro in four moves. [Century Dictionary]
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running (n.)

Old English ærning, "act of one who or that which runs, rapid motion on foot," verbal noun from run (v.). Of a ship, "the action of sailing," 1680s.

Colloquial phrases in (or out) of the running "among (or not among) the lead competitors, competing (or not competing) in a race" (1863) is a metaphor from horse racing, where make the running "set the pace" is recorded from 1837; hence "likely to succeed." Running-shoe is from 1884.

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running (adj.)

"that runs, capable of moving quickly," late 14c., rennynge, present-participle adjective from run (v.), replacing earlier erninde, from Old English eornende. The meaning "rapid, hasty, done on the run" is from c. 1300. The sense of "continuous, carried on continually" is from late 15c.

Running-jump is from 1914. A running-mate (1865) originally was a horse entered in a race to set the pace for another from the same stable who was intended to win; U.S. "vice-presidential candidate" sense is recorded from 1888. Running-board is attested by 1817 in reference to a narrow gangway on either side of a ship or boat; extended by 1907 to the footboards of cars and trucks. 

Running dog is recorded by 1937, from Chinese and later North Korean communist phrases used to describe supposed imperialist lackeys, such as Mandarin zou gou "running dog," on the notion of a dog that runs at its master's command.

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long-running (adj.)

1943, of theatrical productions, from long (adv.) + present participle of run (v.). Related: Longest-running.

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teammate (n.)

also team-mate, 1901, from team (n.) + mate (n.).

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housemate (n.)

also house-mate, 1809, from house (n.) + mate (n.).

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roommate (n.)

also room-mate, "one who shares a room with another or others," 1789, American English, from room (n.) + mate (n.). Short form roomie is by 1918.

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