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- Title
ON THE UPFAMILY EXTENSION OF A DOPPELSEMIGROUP.
- Authors
GAVRYLKIV, V. M.
- Abstract
A family U of non-empty subsets of a set D is called an upfamily if for each set U E U any set F U U belongs to U. The upfamily extension υ(D) of D consists of all upfamilies on D. Any associative binary operation ∗: D×D → D can be extended to an associative binary operation ... In the paper, we show that the upfamily extension (υ(D), -|, -|) of a (strong) doppelsemigroup (D, -|, -|) is a (strong) doppelsemigroup as well and study some properties of this extension. Also we introduce the upfamily functor in the category DSG whose objects are doppelsemigroups and morphisms are doppelsemigroup homomorphisms. We prove that the automorphism group of the upfamily extension of a doppelsemigroup (D, -|, -|) of cardinality |D| ≥ 2 contains a subgroup, isomorphic to C2 × Aut(D, -|, -|). Also we describe the structure of upfamily extensions of all two-element doppelsemigroups and their automorphism groups.
- Subjects
GROUP extensions (Mathematics); SEMIGROUPS (Algebra); BINARY operations; MORPHISMS (Mathematics); HOMOMORPHISMS; AUTOMORPHISM groups
- Publication
Matematychni Studii, 2024, Vol 61, Issue 2, p123
- ISSN
1027-4634
- Publication type
Article
- DOI
10.30970/ms.61.2.123-135