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http://hdl.handle.net/11320/4902
Tytuł: | Stone Lattices |
Autorzy: | Grabowski, Adam |
Słowa kluczowe: | pseudocomplemented lattices Stone lattices Boolean lattices lattice of natural divisors |
Data wydania: | 2015 |
Data dodania: | 16-gru-2016 |
Wydawca: | De Gruyter Open |
Źródło: | Formalized Mathematics, Volume 23, Issue 4, 387–396 |
Abstrakt: | The article continues the formalization of the lattice theory (as structures with two binary operations, not in terms of ordering relations). In the paper, the notion of a pseudocomplement in a lattice is formally introduced in Mizar, and based on this we define the notion of the skeleton and the set of dense elements in a pseudocomplemented lattice, giving the meet-decomposition of arbitrary element of a lattice as the infimum of two elements: one belonging to the skeleton, and the other which is dense.The core of the paper is of course the idea of Stone identity a*⊔a**=T, which is fundamental for us: Stone lattices are those lattices L, which are distributive, bounded, and satisfy Stone identity for all elements a ∈ L. Stone algebras were introduced by Grätzer and Schmidt in [18]. Of course, the pseudocomplement is unique (if exists), so in a pseudcomplemented lattice we defined a * as the Mizar functor (unary operation mapping every element to its pseudocomplement). In Section 2 we prove formally a collection of ordinary properties of pseudocomplemented lattices.All Boolean lattices are Stone, and a natural example of the lattice which is Stone, but not Boolean, is the lattice of all natural divisors of p 2 for arbitrary prime number p (Section 6). At the end we formalize the notion of the Stone lattice B [2] (of pairs of elements a, b of B such that a ⩽ b) constructed as a sublattice of B 2, where B is arbitrary Boolean algebra (and we describe skeleton and the set of dense elements in such lattices). In a natural way, we deal with Cartesian product of pseudocomplemented lattices.Our formalization was inspired by [17], and is an important step in formalizing Jouni Järvinen Lattice theory for rough sets [19], so it follows rather the latter paper. We deal essentially with Section 4.3, pages 423–426. The description of handling complemented structures in Mizar [6] can be found in [12]. The current article together with [15] establishes the formal background for algebraic structures which are important for [10], [16] by means of mechanisms of merging theories as described in [11]. |
Afiliacja: | Institute of Informatics, University of Białystok, Ciołkowskiego 1M, 15-245 Białystok, Poland |
URI: | http://hdl.handle.net/11320/4902 |
DOI: | 10.1515/forma-2015-0031 |
ISSN: | 1426-2630 1898-9934 |
Typ Dokumentu: | Article |
Występuje w kolekcji(ach): | Artykuły naukowe (WInf) Formalized Mathematics, 2015, Volume 23, Issue 4 |
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