The variety of complemented lattices where the Sasaki operations form an adjoint pair
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915073344143360 |
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| author | Cenker, Václav Chajda, Ivan Länger, Helmut |
| author_facet | Cenker, Václav Chajda, Ivan Länger, Helmut |
| contents | The Sasaki projection was introduced as a mapping from the lattice of closed subspaces of a Hilbert space onto one of its segments. To use this projection and its dual so-called Sasaki operations were introduced by the second two authors. In a previous paper there are described several classes of lattices, $λ$-lattices and semirings where the Sasaki operations form an adjoint pair. In the present paper we prove that the class of complemented lattices with this property forms a variety and we explicitly state its defining identities. Moreover, we prove that this variety V is congruence permutable and regular. Hence every ideal I of some member L of V is a kernel of some congruence on L. Finally, we determine a finite basis of so-called ideal terms and describe the congruence $Θ_I$ determined by the ideal I. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_15764 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The variety of complemented lattices where the Sasaki operations form an adjoint pair Cenker, Václav Chajda, Ivan Länger, Helmut Logic 06C15, 06B05, 06C20 The Sasaki projection was introduced as a mapping from the lattice of closed subspaces of a Hilbert space onto one of its segments. To use this projection and its dual so-called Sasaki operations were introduced by the second two authors. In a previous paper there are described several classes of lattices, $λ$-lattices and semirings where the Sasaki operations form an adjoint pair. In the present paper we prove that the class of complemented lattices with this property forms a variety and we explicitly state its defining identities. Moreover, we prove that this variety V is congruence permutable and regular. Hence every ideal I of some member L of V is a kernel of some congruence on L. Finally, we determine a finite basis of so-called ideal terms and describe the congruence $Θ_I$ determined by the ideal I. |
| title | The variety of complemented lattices where the Sasaki operations form an adjoint pair |
| topic | Logic 06C15, 06B05, 06C20 |
| url | https://arxiv.org/abs/2412.15764 |