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).
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Communicated by Lee See Keong.
This work was supported by Grant VEGA 1/0063/14.
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Farkasová, Z., Jakubíková-Studenovská, D. The Lattice of Convex Subsets of a Monounary Algebra. Bull. Malays. Math. Sci. Soc. 40, 583–597 (2017). https://doi.org/10.1007/s40840-017-0456-1
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DOI: https://doi.org/10.1007/s40840-017-0456-1