On the upfamily extension of a doppelsemigroup

Анотація

A family $\mathcal{U}$ of non-empty subsets of a set $D$ is called  an 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 $$*:\upsilon(D)\times \upsilon(D)\to \upsilon(D), \ \ \ \mathcal U*\mathcal V=\big\langle\bigcup_{a\in
U}a*V_a:U\in\mathcal U,\;\;\{V_a\}_{a\in U}\subset\mathcal V\big\rangle.$$
In the paper, we show that the upfamily extension $(\upsilon(D),\dashv,\vdash)$ of a (strong) doppelsemigroup  $(D,\dashv,\vdash)$ is a (strong) doppelsemigroup as well and study some properties of this extension. Also we introduce the upfamily functor in the category $\mathbf {DSG}$ whose objects are doppelsemigroups and  morphisms are doppelsemigroup homomorphisms. We prove that the automorphism group of the upfamily extension of a doppelsemigroup $(D,\dashv, \vdash)$ of cardinality $|D|\geq 2$ contains a subgroup, isomorphic to $C_2\times \mathrm{Aut\mkern 2mu}(D,\dashv, \vdash)$. Also we describe the structure of upfamily extensions of all two-element doppelsemigroups and their automorphism groups.