On Free ω-Continuous and Regular Ordered Algebras

Let E be a set of inequalities between finite Σ-terms. Let Vω and Vr denote the varieties of all ω-continuous ordered Σ-algebras and regular ordered Σ-algebras satisfying E, respectively. We prove that the free Vr-algebra R(X) on generators X is the subalgebra of the corresponding free Vω-algebra Fω(X) determined by those elements of Fω(X) denoted by the regular Σ-coterms. We actually establish this fact as a special case of a more general construction for families of algebras specified by syntactically restricted completeness and continuity properties. Thus our result is also applicable to ordered regular algebras of higher order. 1. ω-Continuous Algebras Let Σ be a ranked alphabet, which will be fixed throughout. A Σ-algebra A is called ordered if A is partially ordered by a relation ≤ with least element – and the algebraic operations are monotone with respect to ≤; that is, if f ∈ Σn and ai, bi ∈ A with ai ≤ bi for 1 ≤ i ≤ n, then f(a1, . . . , an) ≤ f(b1, . . . , bn). A morphism h ∶ A→ B of ordered algebras is a strict monotone map that commutes with the algebraic operations: a ≤ b⇒ h(a) ≤ h(b) h(–) = – h(f(a1, . . . , an)) = f(h(a1), . . . , h(an)) for all a, b, a1, . . . , an ∈ A and f ∈ Σn, n ≥ 0. For each set X, there is a free ordered algebra TX freely generated by X. The elements of TX are represented by the finite partial Σ-terms over X; here partial means that some subterms may be missing, which is the same as having the empty term – in that position. When t ∈ TX and A is an ordered algebra, t induces a function t ∶ A → A in the usual way. An ordered algebra A is ω-continuous [5, 7, 8] if it is ω-complete and the operations are ω-continuous. That is, any countable directed set (or countable chain) C has a supremum ⋁C, and when n ≥ 0, f ∈ Σn, and Ci is a countable directed set for 1 ≤ i ≤ n, then f(⋁C1, . . . ,⋁Cn) =⋁{f(x1, . . . , xn) ∣ xi ∈ Ci, 1 ≤ i ≤ n}. A morphism of ω-continuous algebras is an ω-continuous ordered algebra morphism. 2012 ACM CCS: •Theory of computation → Algebraic semantics; Algebraic language theory ; Tree languages; Categorical semantics; Regular languages.