Doppelsemigroups of k-linked upfamilies

A family [Formula: see text] of non-empty subsets of a set [Formula: see text] is called an upfamily if for each set [Formula: see text] any set [Formula: see text] belongs to [Formula: see text]. For [Formula: see text], an upfamily [Formula: see text] is called [Formula: see text]-linked if [Formula: see text] for any subfamily [Formula: see text] with [Formula: see text]. The extension [Formula: see text] of [Formula: see text] consists of all [Formula: see text]-linked upfamilies on [Formula: see text]. Any associative binary operation [Formula: see text] can be extended to an associative binary operation [Formula: see text] In the paper, we study the structure of the doppelsemigroups [Formula: see text] of [Formula: see text]-linked upfamilies over doppelsemigroups [Formula: see text]. Also we introduce the [Formula: see text]-linked upfamily functors in the category [Formula: see text] whose objects are doppelsemigroups and morphisms are doppelsemigroup homomorphisms, and show that these functors preserve strong doppelsemigroups, doppelsemigroups with left (right) zero, doppelsemigroups with left (right) identity, left (right) zeros doppelsemigroups. We prove that for each [Formula: see text] the automorphism group of the extension [Formula: see text] of a doppelsemigroup [Formula: see text] contains a subgroup, isomorphic to the automorphism group of [Formula: see text]. Also we describe the structure of the doppelsemigroups of [Formula: see text]-linked upfamilies over two-element doppelsemigroups and their automorphism groups.