The commutative Moufang loops with maximum conditions for subloops

TL;DR

This paper establishes equivalences between maximum subloop conditions and finite generation properties in commutative Moufang loops and their multiplication groups, extending to commutative Moufang ZA-loops.

Contribution

It proves that maximum conditions for subloops are equivalent to finite generation of subloops and subgroups in commutative Moufang loops and ZA-loops.

Findings

Maximum condition for subloops is equivalent to finite generation of subloops.

Equivalent conditions are established for the multiplication group.

Results extend to commutative Moufang ZA-loops.

Abstract

It is proved that the maximum condition for subloops in a commutative Moufang loop $Q$ is equivalent with the conditions of finite generating of different subloops of the loop $Q$ and different subgroups of the multiplication group of the loop $Q$. An analogue equivalence is set for the commutative Moufang $ZA$-loops.