Incomparable Elements in Algebraic Lattices with an Application to AFL-theory

Two special types of algebraic (or compactly generated) lattices, called GG- and GH-lattice, are introduced. These lattices are uncountable, and each element in these lattices (apart from the zero and unit) has an incomparable element. The main results characterize those elements which have a largest incomparable element. Then two particular kinds of algebras, called GG- and GH-algebra, are defined and it is shown that the lattice of subalgebras of a GG-algebra [GH-algebra] is a GG-lattice [GH-lattice]. Finally, some applications to the theory of Abstract Families of Languages (or AFL) are discussed.