Lattices, complete lattices and (complete) lower and upper semilattices are all very similar algebraic structures. The homomorphisms between lattices of the same type and also the Isomorphism Theorems hold for all of these structures, but it would be clumsy to write the same statement for each of these classes separately.
I would like to know if there is a general setting for which each of these classes of lattices are particular examples. Universal algebra seems like the way to go; however, most texts that I have found make no mention of how to view a complete (semi)lattice as an algebra in the sense of universal algebra, and I am unsure whether it is possible at all since arbitrary meets and joins do not constitute finitary operations.
Is there a way to achieve this general setting? Some references would be greatly appreciated.