I am interested in algebras whose subalgebra lattice is supremum-founded. Let us call those algebras small.
A complete lattice $(L, \leq)$ is called supremum-founded, if for any two elements $x < y$ from $L$ there is an element $s \in L$ which is minimal with respect to $s \leq y$, $s \not\leq x$.
Of course, every finite lattice is supremum-founded whence every finite algebra is small. Moreover, it is not very difficult to show that every $1$-locally finite algebra (that is, an algebra whose $1$-generated subalgebras are finite) is small.
Does anybody know some more/bigger classes of examples? Is there a suitable reference?