Central subloop

This article defines a property that can be evaluated for a subloop of a loop| View other such properties
ANALOGY: This is an analogue in algebra loop of a property encountered in group. Specifically, it is a subloop property analogous to the subgroup property: central subgroup
View other analogues of central subgroup | View other analogues in algebra loops of subgroup properties (OR, View as a tabulated list)

Definition

A subloop of an algebra loop is termed a central subloop or central subgroup if it is contained in the center of the loop. In other words, a subloop $S$ of an algebra loop $(L,*)$ if, for all $a \in S$ and $x,y \in L$, we have: $\! x * (y * a) = (x * y) * a = a * (x * y) = (a * x) * y$

Note that any central subloop is an abelian group under the induced multiplication because the multiplication operation on it is associative.

Relation with other properties

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Normal subloop Central factor of a loop|FULL LIST, MORE INFO
Nuclear subloop contained in the nucleus |FULL LIST, MORE INFO