The commutative Moufang loops with minimum conditions for subloops II

TL;DR

This paper establishes equivalences between various minimum conditions for subloops in infinite non-associative commutative Moufang loops and examines their structure, especially focusing on loops with normal infinite non-associative subloops.

Contribution

It proves the equivalence of multiple minimum conditions for subloops in infinite commutative Moufang loops and analyzes the structure of loops with normal infinite non-associative subloops.

Findings

Conditions for subloops are equivalent in infinite non-associative commutative Moufang loops.

Loops with normal infinite non-associative subloops have a specific structural characterization.

Abstract

It is proved that the following conditions are equivalent for an infinite non-associative commutative Moufang loop $Q$: 1) $Q$ satisfies the minimum condition for subloops; 2) if the loop $Q$ contains a centrally solvable subloop of class $s$, then it satisfies the minimum condition for centrally solvable subloops of class $s$; 3) if the loop $Q$ contains a centrally nilpotent subloop of class $n$, then it satisfies the minimum condition for centrally nilpotent subloops of class $n$; 4) $Q$ satisfies the minimum condition for non-invatiant associative subloops. The structure of the commutative Moufang loops, whose infinite non-associative subloops are normal, is examined.