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 [0,1], 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