The commutative inverse semigroup of partial abelian extensions

Abstract

This paper is a new contribution to the partial Galois theory of groups. First, given a unital partial action αG of a finite group G on an algebra S such that S is an -partial Galois extension of and a normal subgroup H of G, we prove that induces a unital partial action of G/H on the subalgebra of invariants of S such that is an -partial Galois extension of Second, assuming that G is abelian, we construct a commutative inverse semigroup whose elements are equivalence classes of -partial abelian extensions of a commutative algebra R. We also prove that there exists a group isomorphism between and T(G, A), where ρ is a congruence on and T(G, A) is the classical Harrison group of the G-isomorphism classes of the abelian extensions of a commutative algebra A. It is shown that the study of reduces to the case where G is cyclic. The set of idempotents of is also investigated.