Alfred Tarski presents a lattice-theoretical fixpoint theorem in this paper, which holds in arbitrary complete lattices. The theorem states that if \( f \) is an increasing function on a complete lattice \( \mathfrak{L} = (A, \leq) \), then the set of all fixpoints of \( f \) is non-empty and the system of fixpoints is a complete lattice. Specifically, the join and meet of the fixpoints are also fixpoints. This theorem has various applications and extensions in different areas of mathematics, including: 1. **Simply Ordered Sets and Real Functions**: The theorem is applied to simply ordered sets and real functions, where it helps in proving the existence of fixpoints for quasi-increasing and quasi-decreasing functions. It also leads to the generalized Weierstrass theorem for continuous real functions. 2. **Boolean Algebras**: The lattice-theoretical fixpoint theorem is used to study Boolean algebras, particularly in the context of homogeneous elements and set-theoretical equivalence. Theorems are derived that describe the properties of homogeneity and provide mean-value, equivalence, and interpolation theorems. 3. **Topology**: The theorem is applied to derivative algebras in topology, leading to the generalized Cantor-Bendixson theorem. This theorem states that every closed element in a complete derivative algebra can be decomposed into a perfect and a scattered part. Additionally, it provides a decomposition of closed elements into two disjoint parts, each with specific properties. The paper also discusses the implications of these results for the isomorphism of cardinal products of Boolean algebras and the study of set-theoretical equivalence. The applications of the lattice-theoretical fixpoint theorem in these areas provide valuable insights and extensions to existing theories.Alfred Tarski presents a lattice-theoretical fixpoint theorem in this paper, which holds in arbitrary complete lattices. The theorem states that if \( f \) is an increasing function on a complete lattice \( \mathfrak{L} = (A, \leq) \), then the set of all fixpoints of \( f \) is non-empty and the system of fixpoints is a complete lattice. Specifically, the join and meet of the fixpoints are also fixpoints. This theorem has various applications and extensions in different areas of mathematics, including: 1. **Simply Ordered Sets and Real Functions**: The theorem is applied to simply ordered sets and real functions, where it helps in proving the existence of fixpoints for quasi-increasing and quasi-decreasing functions. It also leads to the generalized Weierstrass theorem for continuous real functions. 2. **Boolean Algebras**: The lattice-theoretical fixpoint theorem is used to study Boolean algebras, particularly in the context of homogeneous elements and set-theoretical equivalence. Theorems are derived that describe the properties of homogeneity and provide mean-value, equivalence, and interpolation theorems. 3. **Topology**: The theorem is applied to derivative algebras in topology, leading to the generalized Cantor-Bendixson theorem. This theorem states that every closed element in a complete derivative algebra can be decomposed into a perfect and a scattered part. Additionally, it provides a decomposition of closed elements into two disjoint parts, each with specific properties. The paper also discusses the implications of these results for the isomorphism of cardinal products of Boolean algebras and the study of set-theoretical equivalence. The applications of the lattice-theoretical fixpoint theorem in these areas provide valuable insights and extensions to existing theories.