Abstract:A family $\mathcal{U}$ of non-empty subsets of a set $D$ is called an {\em upfamily} if for each set $U\in\mathcal{U}$ any set $F\supset U$ belongs to $\mathcal{U}$. The upfamily extension $\upsilon(D)$ of $D$ consists of all upfamilies on~$D$.Any associative binary operation $* \colon D\times D \to D$ can be extended to an associative binary operation $$*:\upsilon(D)\times \upsilon(D)\to \upsilon(D), \ \ \ \mathcal U*\mathcal V=\big\langle\bigcup_{a\inU}a*V_a:U\in\mathcal U,\;\;\{V_a\}_{a\in U}\subset\mathca… Show more
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