The operator of relative complementation
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912033977401344 |
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| author | Chajda, Ivan Länger, Helmut |
| author_facet | Chajda, Ivan Länger, Helmut |
| contents | By the operator of relative complementation is meant a mapping assigning to every element x of an interval [a,b] of a lattice L the set x^{ab} of all relative complements of x in [a,b]. Of course, if L is relatively complemented then x^{ab} is non-empty for each interval [a,b] and every element x belonging to it. We study the question under what condition a complement of x in L induces a relative complement of x in [a,b] It is well-known that this is the case provided L is modular and complemented. However, we present a more general result. Further, we investigate properties of the operator of relative complementation, in particular in the case when the interval [a,b] is a modular sublattice of L or if it is finite. Moreover, we characterize when the operator of relative complementation is involutive and we show a class of lattices where this identity holds. Finally, we establish sufficient conditions under which two different complements of a given element x of [a,b] induce the same relative complement of x in this interval. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_12291 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The operator of relative complementation Chajda, Ivan Länger, Helmut Rings and Algebras 06C20, 06C15, 06C05, 06A15 By the operator of relative complementation is meant a mapping assigning to every element x of an interval [a,b] of a lattice L the set x^{ab} of all relative complements of x in [a,b]. Of course, if L is relatively complemented then x^{ab} is non-empty for each interval [a,b] and every element x belonging to it. We study the question under what condition a complement of x in L induces a relative complement of x in [a,b] It is well-known that this is the case provided L is modular and complemented. However, we present a more general result. Further, we investigate properties of the operator of relative complementation, in particular in the case when the interval [a,b] is a modular sublattice of L or if it is finite. Moreover, we characterize when the operator of relative complementation is involutive and we show a class of lattices where this identity holds. Finally, we establish sufficient conditions under which two different complements of a given element x of [a,b] induce the same relative complement of x in this interval. |
| title | The operator of relative complementation |
| topic | Rings and Algebras 06C20, 06C15, 06C05, 06A15 |
| url | https://arxiv.org/abs/2409.12291 |