No longer necessities, domestic animals became luxuries.
= Being no longer necessities, domestic animals became luxuries.
现在分词短语作状语。系词be的现在分词短语作状语,being常省略。省略后可视为名词词组作状语。
《剑桥语法》将这种结构称为补充语(supplement),对主语作补充说明。
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家畜不再是必需品,而成了奢侈品。necessities是必需品的意思。