Doppelsemigroups of k-linked upfamilies

A family u of non-empty subsets of a set D is called an upfamily if for each set U is an element of u any set F superset of U belongs to u. For k is an element of N\{1}, an upfamily u is an element of v(S) is called k-linked if boolean AND L not equal empty set for any subfamily L subset of u with |L|<= k. The extension N-k(D) of D consists of all k-linked upfamilies on D. Any associative binary operation * : D x D -> D can be extended to an associative binary operation * : N-k(D) x N-k(D) -> N-k(D),u*V = . In the paper, we study the structure of the doppelsemigroups (N-k(D),(sic),proves) of k-linked upfamilies over doppelsemigroups (D,(sic),proves). Also we introduce the k-linked upfamily functors in the category DSG 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 k is an element of N\{1} the automorphism group of the extension (N-k(D),(sic),proves) of a doppelsemigroup (D,(sic),proves) contains a subgroup, isomorphic to the automorphism group of (D,(sic),proves). Also we describe the structure of the doppelsemigroups of k-linked upfamilies over two-element doppelsemigroups and their automorphism groups.