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 α_G of a finite group G on an algebra S such that S is an ${\alpha }_G$-partial Galois extension of $S^{{\alpha }_G}$ and a normal subgroup H of G, we prove that ${\alpha }_G$ induces a unital partial action ${\alpha }_{G/H}$ of G/H on the subalgebra of invariants $S^{{\alpha }_H}$ of S such that $S^{{\alpha }_H}$ is an ${\alpha }_{G/H}$-partial Galois extension of $S^{{\alpha }_G}.$ Second, assuming that G is abelian, we construct a commutative inverse semigroup $T_{\mathit{par}}(G,R),$ whose elements are equivalence classes of ${\alpha }_G$-partial abelian extensions of a commutative algebra R. We also prove that there exists a group isomorphism between $T_{\mathit{par}}(G,R)/\rho$ and T(G, A), where ρ is a congruence on $T_{\mathit{par}}(G,R)$ 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 $T_{\mathit{par}}(G,R)$ reduces to the case where G is cyclic. The set of idempotents of $T_{\mathit{par}}(G,R)$ is also investigated.

Keywords:

• Harrison group
• inverse semigroup
• partial Galois abelian extension

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 13B05
• 13A50
• Secondary: 20M14
• 20M18