Let $\mathbf{C}$ denote the category of models of an algebraic theory in $\mathbf{Set}.$ Now suppose $X$ is an object of $\mathbf{Set}$. Is there a traditional phrase used to describe those objects of $\mathbf{C}$ that can be expressed as quotients of the algebra freely generated by $X$?
For example, suppose the phrase is "$X$-quotational." Then to say an object of $\mathbf{Grp}$ is $1$-quotational is just to say it is a cyclic group. Further, we can define that an algebraic structure is $\kappa$-dimensional iff firstly, it is $\kappa$-quotational, and secondly, it fails to be $\nu$-quotational for all cardinal numbers $\nu < \kappa.$ (I don't know if this is actually useful though.)
If $X$ has $n$ elements, you talk about $n$-generated objects. This is also true when $X$ is infinite and therefore $n$ is a cardinal.