Every superatomic subalgebra of an interval algebra is embeddable in an ordinal algebra
Uri Abraham, Robert Bonnet · Proceedings of the American Mathematical Society · 1992
Let us recall that a Boolean algebra is superatomic if every subalgebra is atomic. So by the definition, every subalgebra of a superatomic algebra is superatomic. An obvious example of a superatomic algebra is the interval algebra generated by a well-ordered chain. In this work, we show that every superatomic subalgebra of an interval algebra is embeddable in an ordinal algebra, that is by definition, an interval algebra generated by a well-ordered chain. As a corollary, if $B$ is an infinite superatomic subalgebra of an interval algebra, then $B$ and the set $\operatorname {At}(B)$ of atoms of $B$ have the same cardinality.