Geodesic
c-ideals in complemented posets
Mathematica Bohemica, Tome 149 (2024) no. 3, pp. 305-316
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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.
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.
DOI : 10.21136/MB.2023.0108-22
Classification : 06A11, 06C15
Keywords: complemented poset; antitone involution; ideal; filter; ultrafilter; c-ideal; c-filter; c-condition; separation theorem 
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Chajda, Ivan; Kolařík, Miroslav; Länger, Helmut. c-ideals in complemented posets. Mathematica Bohemica, Tome 149 (2024) no. 3, pp. 305-316. doi: 10.21136/MB.2023.0108-22

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