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Old English dæge "female servant, housekeeper, maid," from Proto-Germanic *daigjon (cf. Old Norse deigja "maid, female servant," Swedish deja "dairymaid"), from PIE *dheigh- "to form, build" (see dough). Now obsolete (though OED says, "Still in living use in parts of Scotland"), it forms the first element of dairy and the second of lady.
The ground sense seems to be "kneader, maker of bread;" advancing by Old Norse deigja and Middle English daie to mean "female servant, woman employed in a house or on a farm." Dæge as "servant" is the second element in many surnames ending in -day (e.g. Faraday, and perhaps Doubleday "servant of the Twin," etc.).
1650s, "title of a military commander in Muslim north Africa," from Turkish dai "maternal uncle," a friendly title used of older men, especially by the Janissaries of Algiers of their commanding officers. There were also deys in Tunis and Tripoli.
in the Ottoman provinces of Algiers and Tunis, an honorary title conferred upon exceptionally able corsair leaders; also, a lower rank of officer in the Janissaries. In late 16th-century Tunis, a dey commanded the army and eventually was in sole control of the state, but by 1705 the title had disappeared from official lists. The head of the Algerian regency, elected by fellow Janissary officers (from 1689), was titled dey, and, though his family life was restricted to prevent succession claims and he was confined to Algiers, he had virtually absolute power; 30 such deys ruled Algiers in succession between 1671 and 1830