Algebra in superextensions of groups, II: cancelativity and centers

• Veröffentlicht in: Algebra and Discrete Mathematics
• Datum: 2008
• Hauptverfasser: Banakh, T., Gavrylkiv, V.
• Format: Artikel
• Sprache: Englisch
• Veröffentlicht: Інститут прикладної математики і механіки НАН України 2008
• Online Zugang: https://nasplib.isofts.kiev.ua/handle/123456789/153356
• Назва журналу: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Zitieren: Algebra in superextensions of groups, II: cancelativity and centers / T. Banakh, V. Gavrylkiv // Algebra and Discrete Mathematics. — 2008. — Vol. 7, № 4. — С. 1–14. — Бібліогр.: 10 назв. — англ.

Given a countable group X we study the algebraic structure of its
 superextension λ(X). This is a right-topological semigroup consisting of all maximal linked systems on X endowed with the operation 
 
 A∘B={C⊂X:{x∈X:x−1C∈B}∈A} 
 
 that extends the group operation of X. We show that the subsemigroup λ∘(X) of free maximal linked systems contains an open dense subset of right cancelable elements. Also we prove that the topological center of λ(X) coincides with the subsemigroup λ∙(X) of all maximal linked systems with finite support. This result is applied to show that the algebraic center of λ(X) coincides with the algebraic center of X provided X is countably infinite. On the other hand, for finite groups X of order 3≤|X|≤5 the algebraic center of λ(X) is strictly larger than the algebraic center of X.