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Implication in Weakly and Dually Weakly Orthomodular Lattices

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Algebraic Perspectives on Substructural Logics

Part of the book series: Trends in Logic ((TREN,volume 55))

Abstract

Weakly orthomodular and dually weakly orthomodular lattices were introduced by the authors in a recent paper. Similarly as for orthomodular lattices we try to introduce an implication in these lattices which can be easily axiomatized and which yields a nice lattice structure. As shown in the paper, this can be realized in several different ways. Moreover, we reveal the connection of weakly and dually weakly orthomodular lattices to residuated structures. Furthermore, we provide a characterization of these lattices by means of certain generalized measures.

Support of the research of both authors by the Austrian Science Fund (FWF), project I 4579-N, and the Czech Science Foundation (GACR), project 20-09869L, entitled “The many facets of orthomodularity”, as well as by ÖAD, project CZ 02/2019, entitled “Function algebras and ordered structures related to logic and data fusion”, and, concerning the first author, by IGA, project PrF 2020 014, is gratefully acknowledged.

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Acknowledgements

Support of the research of the authors by the Austrian Science Fund (FWF), project I 4579-N, and the Czech Science Foundation (GACR), project 20-09869L, entitled “The many facets of orthomodularity”, as well as by ÖAD, project CZ 02/2019, entitled “Function algebras and ordered structures related to logic and data fusion”, and, concerning the first author, by IGA, project PrF 2020 014, is gratefully acknowledged. The authors thank the anonymous referee for his/her careful checking and for many valuable suggestions.

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Correspondence to Ivan Chajda .

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Chajda, I., Länger, H. (2021). Implication in Weakly and Dually Weakly Orthomodular Lattices. In: Fazio, D., Ledda, A., Paoli, F. (eds) Algebraic Perspectives on Substructural Logics. Trends in Logic, vol 55. Springer, Cham. https://doi.org/10.1007/978-3-030-52163-9_4

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