The Lattice of Convex Subsets of a Monounary Algebra

Abstract

We deal with the lattice \(\mathrm {Co}(A,f)\) of all convex subsets of a monounary algebra (A, f). Monounary algebras (A, f) with the property that the lattice \(\mathrm {Co}(A,f)\) is distributive, modular, semimodular, selfdual, complemented, respectively, are characterized. For algebras possessing no cycles with more than two elements, the properties distributive, modular and selfdual are equivalent. Moreover, the lattice \(\mathrm {Co}(A,f)\) is modular iff it is selfdual, and then the distributive lattice \(\mathrm {Co}(A,f)\) is equal to the lattice \({\mathcal {P}}(A)\) (power set of A). Further, we find conditions under which a distributive (modular, etc.) lattice L is representable as the lattice \(\mathrm {Co}(A,f)\) for some monounary algebra (A, f).