Superextensions of doppelsemigroups

Abstract

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 $\mathcal{U}$. An upfamily $\mathcal L$ of subsets of $D$ is said to be linked if $A\cap B\ne\emptyset$ for all $A,B\in\mathcal L$. A linked upfamily $\mathcal M$ of subsets of $D$ is maximal linked if $\mathcal M$ coincides with each linked upfamily $\mathcal L$ on $D$ that contains $\mathcal M$. The superextension $\lambda(D)$ of $D$ consists of all maximal linked upfamilies on $D$. Any associative binary operation $* : D\times D \to D$ can be extended to an associative binary operation $$*:\lambda(D)\times \lambda(D)\to \lambda(D), \quad \mathcal M*\mathcal L=\Big\langle\bigcup_{a\in M}a*L_a:M\in\mathcal M,\;\{L_a\}_{a\in M}\subset\mathcal L\Big\rangle.$$ In the paper, we investigate the structure of the doppelsemigroup $(\lambda(D),\dashv,\vdash)$ of maximal linked upfamilies on a doppelsemigroup $(D,\dashv,\vdash)$. In particular, we study right and left zeros and identities, commutativity, the center, ideals of the superextension $(\lambda(D),\dashv,\vdash)$ of a doppelsemigroup $(D,\dashv, \vdash)$. We introduce the superextension functor $\lambda$ in the category $\mathbf {DSG}$, whose objects are doppelsemigroups and morphisms are doppelsemigroup homomorphisms, and show that this functor preserves strong doppelsemigroups, doppelsemigroups with left (right) zero, doppelsemigroups with left (right) identity, left (right) zeros doppelsemigroups. Also we prove that the automorphism group of the superextension of a doppelsemigroup $(D,\dashv, \vdash)$ contain a subgroup, isomorphic to the automorphism group of $(D,\dashv, \vdash)$.