Let $E$ be a monoid. I know that the intersection of any family of submonoids of $E$ is again a submonoid of $E$. Under what conditions is the union of an arbitrary family of submonoids of $E$ a submonoid?
2026-03-25 19:05:37.1774465537
Union of submonoids of a monoid
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If you have a family of submonoids $\left\lbrace E_i\right\rbrace_i,\ E_i\subset E,$ I suppose that you need to ask for the condition that for $x\in E_1,\ y\in E_2; x,y\notin E_1\cap E_2 ,$ you have that $x*y\in E_1\cup E_2.$