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Formalized Mathematics

(a computer assisted approach)

Editor-in-Chief: Matuszewski, Roman

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SCImago Journal Rank (SJR) 2016: 0.207
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Volume 22, Issue 3


Formalization of Generalized Almost Distributive Lattices

Adam Grabowski
Published Online: 2014-03-31 | DOI: https://doi.org/10.2478/forma-2014-0026


Almost Distributive Lattices (ADL) are structures defined by Swamy and Rao [14] as a common abstraction of some generalizations of the Boolean algebra. In our paper, we deal with a certain further generalization of ADLs, namely the Generalized Almost Distributive Lattices (GADL). Our main aim was to give the formal counterpart of this structure and we succeeded formalizing all items from the Section 3 of Rao et al.’s paper [13]. Essentially among GADLs we can find structures which are neither V-commutative nor Λ-commutative (resp., Λ-commutative); consequently not all forms of absorption identities hold.

We characterized some necessary and sufficient conditions for commutativity and distributivity, we also defined the class of GADLs with zero element. We tried to use as much attributes and cluster registrations as possible, hence many identities are expressed in terms of adjectives; also some generalizations of wellknown notions from lattice theory [11] formalized within the Mizar Mathematical Library were proposed. Finally, some important examples from Rao’s paper were introduced. We construct the example of GADL which is not an ADL. Mechanization of proofs in this specific area could be a good starting point towards further generalization of lattice theory [10] with the help of automated theorem provers [8].

Keywords : almost distributive lattices; generalized almost distributive lattices; lattice identities


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About the article

Received: 2014-09-26

Published Online: 2014-03-31

Published in Print: 2014-09-01

Citation Information: Formalized Mathematics, Volume 22, Issue 3, Pages 257–267, ISSN (Online) 1898-9934, DOI: https://doi.org/10.2478/forma-2014-0026.

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© by Adam Grabowski. This work is licensed under the Creative Commons Attribution-ShareAlike 3.0 License. (CC BY-SA 3.0) BY-SA 3.0

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