Example. $\omega$ -algebraic cpos [dt-0042]

Liam O'Connor

Many standard examples of cpos are $\omega$ -algebraic cpos: finite cpos, subsets $\mathcal {P}(X)$ of a set $X$ , partial functions $\mathbb {N} \nrightarrow \mathbb {N}$ and submonoids of a monoid. In general, any cpo of subalgebras (e.g. subgroups of a group, subrings of a ring etc.) is $\omega$ -algebraic. This is the origin of the terminology.

Example. $\omega$ -algebraic cpos [dt-0042]

Definition. Algebraicity [dt-0041]

Example. $\omega$ -algebraic cpos [dt-0042]

Counterexample. Non-algebraic cpo [dt-0043]

Theorem. Nothing suddenly invented at infinity [dt-0045]

Proof.

Exercise. [dt-0044]

Solution.

Definition. Down-closure [dt-0047]

Definition. Ideal [dt-0048]

Definition. Ideal completion [dt-0046]

Theorem. Representation theorem for algebraic cpos [dt-0049]