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Trump posted social media material suggesting New Mexico should be renamed New America.
The reporting says he shared an altered map or video showing the state name changed, and one account says he cited prestige and beauty.
One report says he gave no explanation, leaving the intent unclear, while another says some media reports traced the idea to an internet hoax.
New Mexico Governor Michelle Lujan Grisham rejected the proposal and said the name is not up for debate.
More context
She also accused the President of trying to distract Americans from issues like soaring gas prices, and Representative Gabe Vasquez also rejected the idea.
One report says a sitting president has no constitutional power to unilaterally rename a U.S. state.
The story also notes this fits with Trump’s earlier efforts to rename geographical landmarks.