May be this is too naive, but for people accustomed to more or less conventional algebraic objects (like groups, Lie and associative algebras, etc.), it could be revealing to know that there are universal algebras without proper subalgebras (for one such class of algebars, see, e.g., K. Kaarli, Subalgebras of the squares of weakly diagonal majority algebras, Studia Sci. Math. Hungar. 49 (2012), N4, 509-524)

Free semigroups can be embedded into metabelian groups (see, e.g. G. Bergman, Hyperidentities of groups and semigroups, Aequat. Math. 22 (1981), 315-317).