# Normality is not upper join-closed for algebra loops

From Groupprops

ANALOGY BREAKDOWN: This is the breakdown of the analogue in algebra loops of a fact encountered in group. The old fact is: normality is upper join-closed.

View other analogue breakdowns of normality is upper join-closed|View other analogue breakdowns from group to algebra loop

## Statement

It is possible to have the following: an algebra loop , a subloop of , two subloops of containing , such that is a normal subloop in both and but is not a normal subloop of the subloop generated by and .

In fact, we can construct a counterexample that is an Alternative loop (?), and in fact, is a Moufang loop (?).

## Related facts

- Normality is upper join-closed
- Normality satisfies intermediate subloop condition
- Ideal property is upper join-closed for Lie rings
- Ideal property is not upper join-closed for alternating rings

## Proof

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