Definition 9

Let $E$ be a magma and $X$ a subset of $E$. The set of elements of $E$ which commute with each of the elements of $X$ is called the centralizer of $X$.

Let $X$ and $Y$ be two subsets of $E$ and $X'$ and $Y'$ their respective centralizers. If $X \subset Y$, then $Y' \subset X'$.

Let $(X_i)_{i \in I}$ be a family of subsets of $E$ and for all $i \in I$ let $X'_i$ be the centralizer of $X_i$. The centralizer of $\bigcup_{i \in I} X_i$ is $\bigcap_{i \in I} X'_i$.

Let $X$ be a subset of $E$ and $X'$ the centralizer of $X$. The centralizer $X''$ of $X'$ is called the bicentralizer of $X$. Then $X \subset X''$. The centralizer $X'''$ of $X''$ is equal to $X'$. For $X'$ is contained in its bicentralizer $X'''$ and the relation $X \subset X''$ implies $X'''' \subset X'$.