Algebraic lattice

compactly-generated lattice

A lattice each element of which is the union (i.e. the least upper bound) of some set of compact elements (cf. Compact lattice element). A lattice is isomorphic to the lattice of all subalgebras of some universal algebra if and only if it is both complete and algebraic. These conditions are also necessary and sufficient for the lattice to be isomorphic to the congruence lattice of some universal algebra (the Grätzer–Schmidt theorem). In both cases it is assumed that the arity of the operations of the universal algebra is finite.

References

[1]	G. Birkhoff, "Lattice theory" , Colloq. Publ. , 25 , Amer. Math. Soc. (1967)
[2]	G. Grätzer, "General lattice theory" , Birkhäuser (1978) (Original: Lattice theory. First concepts and distributive lattices. Freeman, 1978)

Comments

Algebraic lattices are an important special case of continuous lattices, for which see [a1]. The original reference to the Grätzer–Schmidt theorem is [a2].

References

[a1]	G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M.V. Mislove, D.S. Scott, "A compendium of continuous lattices" , Springer (1980) ISBN 3-540-10111-X MR0614752 Zbl 0452.06001
[a2]	G. Grätzer, E.T. Schmidt, "Characterizations of congruence lattices of abstract algebras" Acta. Sci. Math. (Szeged) , 24 (1963) pp. 34–59