bounded lattice

bounded lattice (plural bounded lattices)

1. (algebra, order theory) 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, LNAI 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, LNAI 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 , or, more generally, of a bounded lattice ${\displaystyle {\mathcal {L}}}$.

   • 2018, Gül Deniz Çayli, Funda Karaçal, “Some Remarks on Idempotent Nullnorms on Bounded Lattices”, in Vicenç Torra, Radko Mesiar, Bernard De Baets, editors, Aggregation Functions in Theory and in Practice, Springer,, page 32:

     In this paper, we study idempotent nullnorms on bounded lattices. We prove that there is no idempotent nullnorm on a distributive bounded lattice L different from the proposal in [13].

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, ${\displaystyle (L,\lor ,\land ,0,1)}$. Regarding ${\displaystyle L}$ as an already defined lattice leads to the join and meet functions being, implicitly, defined in terms of the partial relation, ${\displaystyle \leq }$. Alternatively (regarding ${\displaystyle L}$ as a set), the partial relation can be defined in terms of the join and meet functions.

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

• lower bounded lattice
• upper bounded lattice

• Lattice (order) on Wikipedia.Wikipedia
• Greatest and least elements on Wikipedia.Wikipedia
• Heyting algebra on Wikipedia.Wikipedia