Compact Element - Algebraic Posets

Algebraic Posets

A poset in which every element is the supremum of the compact elements below it is called an algebraic poset. Such posets which are dcpos are much used in domain theory.

As an important special case, an algebraic lattice is a complete lattice L, such that every element x of L is the supremum of the compact elements below x.

A typical example (which served as the motivation for the name "algebraic") is the following:

For any algebra A (for example, a group, a ring, a field, a lattice, etc.; or even a mere set without any operations), let Sub(A) be the set of all substructures of A, i.e., of all subsets of A which are closed under all operations of A (group addition, ring addition and multiplication, etc.) Here the notion of substructure includes the empty substructure in case the algebra A has no nullary operations.

Then:

• The set Sub(A), ordered by set inclusion, is a lattice.
• The greatest element of Sub(A) is the set A itself.
• For any S, T in Sub(A), the greatest lower bound of S and T is the set theoretic intersection of S and T; the smallest upper bound is the subalgebra generated by the union of S and T.
• The set Sub(A) is even a complete lattice. The greatest lower bound of any family of substructures is their interesction.
• The compact elements of Sub(A) are exactly the finitely generated substructures of A.
• Every substructure is the union of its finitely generated substructures; hence Sub(A) is an algebraic lattice.

Also, a kind of converse holds: Every algebraic lattice is isomorphic to Sub(A) for some algebra A.

There is another algebraic lattice which plays an important role in universal algebra: For every algebra A we let Con(A) be the set of all congruence relations on A. Each congruence on A is a subalgebra of the product algebra AxA, so Con(A) ⊆ Sub(AxA). Again we have

• Con(A), ordered by set inclusion, is a lattice.
• The greatest element of Con(A) is the set AxA, which is the congruence corresponding to the constant homomorphism. The smallest congruence is the diagonal of AxA, corresponding to isomorphisms.
• Con(A) is a complete lattice.
• The compact elements of Con(A) are exactly the finitely generated congruences.
• Con(A) is an algebraic lattice.

Again there is a converse: By a theorem of G. Grätzer and E.T.Schmidt, every algebraic lattice is isomorphic to Con(A) for some algebra A.