Bounded lattice

English

Noun

1. Template:Lb Any lattice (type of partially ordered set) that has both a greatest and a least element.
   • 2004, Anna Maria Radzikowska, Etienne Kerre, On L-Fuzzy Rough Sets, Leszek Rutkowski, Jörg Siekmann, Ryszard Tadeusiewicz, Lotfi A. Zadeh (editors), Artificial Intelligence and Soft Computing — ICAISC 2004: 7th International Conference, Proceedings, Springer, Template:W 3070, page 526,

     A residuated lattice is an extension of a bounded lattice by a monoid operation and its residuum, which are abstract counterparts of a triangular norm and a fuzzy residual implication, respectively.

   • 2006, Bart Van Gasse, Chris Cornelis, Glad Deschrijver, Etienne Kerre, Triangle Lattices: Towards an Axiomatization of Interval-Valued Residuated Lattices, Salavatore Greco, Yukata Hata, Shoji Hirano, Masahiro Inuiguchi, Sadaaki Miyamoto, Hung Son Nguyen, Roman Słowiński (editors), Rough Sets and Current Trends in Computing: 5th International Conference, Proceedings, Springer, Template:W 4259, page 117,

     Indeed, in the scope of these logics, formulas can be assigned not only 0 and 1 as truth values, but also elements of [0,1], or, more generally, of a bounded lattice $ℒ$.

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Usage notes

The greatest element is usually denoted 1 and serves as the identity element of the meet operation, ∧. The least element, usually denoted 0, serves as the identity element of the join operation, ∨. The notations ⊤ and ⊥ are also used, less often, for greatest and least element respectively.

A bounded lattice may be defined formally as a tuple, $(L,\vee ,\wedge ,0,1)$. Regarding $L$ as an already defined lattice leads to the join and meet functions being, implicitly, defined in terms of the partial relation, $\le$. Alternatively (regarding $L$ as a set), the partial relation can be defined in terms of the join and meet functions.

For any $x\in L,0\le x\le 1$. That is, the elements 0 and 1 are each comparable with every other element of the lattice.

Derived terms

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Further reading

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