Zur Vollständigkeit der Induktiven Gruppoide der Partiellen Automorphismen von Algebren

Abstract

For any algebraA letГ(A) be the set of partial automorphisms (isomorphisms between subalgebras). With the natural multiplication it is an inductive groupoid in the sense of Ehresmann.Г(A) is complete iff every subset ofГ(A) which is compatible with the semi-ordering has an upper bound. The fact, whetherГ(A) is complete or not, depends on the defining operations ofA. For every direct familyF = (Λ, (A λ ) λ∈Λ ,(ϕλμ)λ, μ∈ Λ,λ ≤ μ of algebras such that allϕ λμ,λ, μ ∈ Δ withλ ≤ μ are one-to-one functions, the direct limit is complete iff allГ(Aλ) are complete. We give some theorems on the decomposition of inductive groupoids, and employ them in proving the completeness ofГ(A) to variousA. In particular, we obtain that, in case whenG is a finite group,Г(G) is complete iffG is either cyclic or direct product of a noncyclic group of order 4 and a cyclic group of odd order. For finite acyclic ringsR and finite fieldsK the inductive groupoidsГ(R) andГ(K) are complete. Further we deal with the question, to what extent algebras are determined by their inductive groupoids. (An algebraA of a classS is defined byГ(A) iff, for any algebraB of the classS, isomorphism betweenГ(A) andГ(B) implies isomorphism betweenA andB.) Particular attention is paid to finite groups. In general, algebras of classesS are not defined within the classS by their inductive groupoids.