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Lattices and Boolean Algebra
Published in B. V. Senthil Kumar, Hemen Dutta, Discrete Mathematical Structures, 2019
B. V. Senthil Kumar, Hemen Dutta
Definition 5.4.4 Distributive Lattice:A lattice 〈L, ⋆, ⊕〉 is called a distributive lattice if for any a, b, c ∈ L,a⋆(b⊕c)=(a⋆b)⊕(a⋆c)a⊕(b⋆c)=(a⊕b)⋆(a⊕c).
Devising and Synthesis of Mems and Nems
Published in Sergey Edward Lyshevski, Mems and Nems, 2018
A lattice is a partially ordered set where for any pair of sets (hypotheses) there is a least upper bound and greatest lower bound. Let our current hypothesis is H1 and the current training example is H2. If H2 is a subset of H1, then no change of H1 is required. If H2 is not a subset of H1, then H1 must be changed. The minimal generalization of H1 is the least upper bound of H2 and H1, and the minimal specialization of H1 is the greatest lower bound of H2 and H1. Thus, the lattice serves as a map that allows us to locate the current hypothesis H1 with reference to the new information H2. There exists the correspondence between the algebra of propositional logic and the algebra of sets. We refer to a hypotheses as logical expressions, as rules that define a concept, or as subsets of the possible instances constructible from some set of dimensions. Furthermore, union and intersection were the important operators used to define a lattice. In addition, the propositional logic expressions can also be organized into a corresponding lattice to implement the artificial learning.
Devising and Synthesis of NEMS and MEMS
Published in Sergey Edward Lyshevski, Nano- and Micro-Electromechanical Systems, 2018
A lattice is a partially ordered set where for any pair of sets (hypotheses) there is a least upper bound and greatest lower bound. Let our current hypothesis be H1 and the current training example be H2. If H2 is a subset of H1, then no change of H1 is required. If H2 is not a subset of H1, then H1 must be changed. The minimal generalization of H1 is the least upper bound of H2 and H1, and the minimal specialization of H1 is the greatest lower bound of H2 and H1. Thus, the lattice serves as a map that allows us to locate the current hypothesis H1 with reference to the new information H2. There exists the correspondence between the algebra of propositional logic and the algebra of sets. We refer to hypotheses as logical expressions, as rules that define a concept, or as subsets of the possible instances constructible from some set of dimensions. Furthermore, union and intersection were the important operators used to define a lattice. In addition, the propositional logic expressions can also be organized into a corresponding lattice to implement the artificial learning.
Some further construction methods for uninorms on bounded lattices
Published in International Journal of General Systems, 2023
Gül Deniz Çaylı, Ümit Ertuğrul, Funda Karaçal
A lattice is a partially ordered set satisfying that each two elements have a greatest lower bound, called infimum and denoted as , as well as a smallest upper bound, called supremum and denoted by . A lattice is called bounded if it has a top element and a bottom element, written as 1 and 0, respectively. For , the symbol x<y means that and . The elements x and y in L are comparable if or . Otherwise, x and y are called incomparable, in this case, we use the notation .
New construction approaches of uninorms on bounded lattices
Published in International Journal of General Systems, 2021
A lattice (Birkhoff 1967) is a nonempty set L equipped with a partial order ≤ such that any two elements x and y have a greatest lower bound (called meet or infimum), denoted by , as well as a smallest upper bound (called join or supremum), denoted by . For , the symbol a<b means that and . The elements a and b in L are comparable if or b<a. Otherwise, a and b are incomparable, in this case we use the notation . For the fixed element a in L, the set of all with will be denoted by i.e. The transpose of a partial order ≤ of a lattice L, denoted by ≥, i.e. if and only if , is also a partial order on L and the meet and the join with respect to ≥ are ∨ and ∧, respectively. That is to say, is also a lattice, called the dual lattice of .
Generalizing expected values to the case of L *-fuzzy events
Published in International Journal of General Systems, 2021
Erich Peter Klement, Fateme Kouchakinejad, Debashree Guha, Radko Mesiar
Observe that the lattices and are isomorphic, i.e. have the same mathematical structure, but their semantics may be quite different. The values of the membership function of an -fuzzy set are typically given by a pair of numbers representing the degree of membership and the degree of non-membership of the object x in the -fuzzy set A. In the case of interval-valued fuzzy sets (based on ), the values of its membership function are usually identified with , i.e. with a subinterval of (in this sense interval-valued fuzzy sets are special examples of fuzzy set of type 2 – for a discussion of these generalizations of fuzzy sets, see, e.g. Zadeh 1975 and Walker and Walker 2009).