When I was studying the following lemma from an article , I faced to some notations I was not familiar with them, I would appreciate your help to find out them:
Lemma:Let $X$ be an inductive class of algebras over a field $K$. Suppose $X$ is closed under subalgebra and $L \in X$. Then there exists an algebra $H \in X$ containing $L$ such that its dimension is at most $$ max \{ \aleph_{0} , dim (L), |k|\}$$. What is the meaning of "inductive class" and "close under subalgebra"?
The order relation in this class X is the existence of monomorphisms, so $X$ is inductive , this means that any monomorphism chain of algebra $ A_0\hookrightarrow A_1\hookrightarrow\cdot \cdot \cdot \hookrightarrow A_n$ has a limit in $X$ (that is inductive limit of this chain).