Existentially closed semilattices (Q1083457)

An algebra A is algebraically closed (existentially closed) in a class K of algebras, if every finite system of equations (and inequalities) with constants from A which is solvable in a K-extension of A already has a solution in A. In the paper under review these properties are investigated for the class BSL of bounded semilattices. In the first part it is shown that BSL has the strong amalgamation property: it follows from an earlier theorem by the author [Arch. Math. Logik Grundlagenforsch. 19, 23-29 (1978; Zbl 0408.03024)] that the class BSL of existentially closed bounded semilattices is the model completion of BSL and that it is \(\aleph_ 0\)-categorical. In the second part, an axiomatic description of BSL' and the class BSL'' of algebraically closed bounded semilattices is given; furthermore, some basic properties of these classes and some connections among these and related classes of lattices are exhibited. Throughout the article, a fruitful mixture of algebraic and model theoretic techniques is employed.