Plural of asherite; people who worship or are devoted to the goddess Asherah, or members of a tribe associated with ash or ash-trees.
From asherah or ash, plus -ite, a suffix forming nouns meaning 'people who belong to or practice something,' similar to how 'Israelite' means a member of Israel.
Using '-ites' to label religious groups was common in ancient times—'Asherites' might refer to Asherah's worshippers, but it also shows how identity, religion, and geography were often expressed through the same grammatical structure in ancient languages.
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