## B-combinatory algebras.(English)Zbl 0654.03034

This work introduces a class of algebras generalizing both the applicative systems of the author’s earlier paper [C. R. Acad. Bulg. Sci. 37, 561-564 (1984; Zbl 0544.03020)] and the operative spaces studied in the reviewer’s book [Algebraic recursion theory (1986; Zbl 0613.03018)]. Namely, a B-combinatory algebra is a poset $${\mathcal F}$$ augmented with fixed elements: A, C, D, O, I, $$D\check{\;}$$, J, E, $$E_ 0$$, $$E_ 1$$, T, F, and monotonic operations: multiplication $$\lambda\phi\psi\cdot \phi\psi$$ and branching $$\lambda$$ $$\phi$$ $$\psi$$ $$\cdot (\phi,\psi)$$ such that the following axioms hold: $$((A\phi)\psi)\chi =\phi (\psi \chi)$$, $$(C\phi)((D\psi)\chi)=(\phi \psi)\chi$$, $$0\leq \phi$$, $$\phi 0=0$$, $$I\phi =\phi$$, $$(C\phi)(D \check{\;}\psi)=\phi \psi$$, $$(C\chi)(J(\phi,\psi))=(\chi \phi,\chi \psi)$$, $$((E_ 0\chi)\phi,(E_ 1\chi)\psi)E=\chi (\phi,\psi)$$, $$(\phi,\psi)T=\phi$$ and $$(\phi,\psi)F=\psi$$. The central result is an abstract version of the First Recursion Theorem.
Reviewer: L.Ivanov

### MSC:

 03D75 Abstract and axiomatic computability and recursion theory

### Keywords:

B-combinatory algebra

### Citations:

Zbl 0544.03020; Zbl 0613.03018