Received August 2, 2022. Published online June 28, 2023.
Abstract: In their recent paper on posets with a pseudocomplementation denoted by $*$ the first and the third author introduced the concept of a $*$-ideal. This concept is in fact an extension of a similar concept introduced in distributive pseudocomplemented lattices and semilattices by several authors, see References. Now we apply this concept of a c-ideal (dually, c-filter) to complemented posets where the complementation need neither be antitone nor an involution, but still satisfies some weak conditions. We show when an ideal or filter in such a poset is a c-ideal or c-filter, and we prove basic properties of them. Finally, we prove the so-called separation theorems for c-ideals. The text is illustrated by several examples.
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Affiliations: Ivan Chajda, Palacký University Olomouc, Faculty of Science, Department of Algebra and Geometry, 17. listopadu 12, 771 46 Olomouc, Czech Republic, e-mail: ivan.chajda@upol.cz; Miroslav Kolařík, Palacký University Olomouc, Faculty of Science, Department of Computer Science, 17. listopadu 12, 771 46 Olomouc, Czech Republic, e-mail: miroslav.kolarik@upol.cz; Helmut Länger (corresponding author), TU Wien, Fakultät für Mathematik und Geoinformation, Institut für Diskrete Mathematik und Geometrie, Wiedner Hauptstrasse 8-10, 1040 Wien, Austria, and Palacký University Olomouc, Faculty of Science, Department of Algebra and Geometry, 17. listopadu 12, 771 46 Olomouc, Czech Republic, e-mail: helmut.laenger@tuwien.ac.at
