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Mathematical Morphology with Noncommutative Symmetry Groups
Published in Edward R. Dougherty, Mathematical Morphology in Image Processing, 2018
A complete lattice (ℒ, ≤) is a partially ordered set ℒ with order relation ≤, a supremum or join operation written ∨ and an infimum or meet operation written ∧, such that every (finite or infinite) subset of ℒ has a supremum (smallest upper bound) and an infimum (greatest lower bound). In particular, there exist two universal bounds, the least element written Oℒ and the greatest element Iℒ In the case of the power lattice P(E) of all subsets of a set E, the order relation is setinclusion ⊆, the supremum is the union ∪ of sets, the infimum is the intersection ∩ of sets, the least element is the empty set 0, and the greatest element is the set e itself. An atom is an element X of a lattice ℒ such that for any Y ε, 0 ℒ ≤ Y ≤ Ximplies that Y = Oℒ or Y = X. A complete lattice ℒ is called atomic if every element of ℒ is the supremum of the atoms less than or equal to it. It is called Boolean if (1) it satisfies the distributivity laws X ∨ (Y ∧ Z) = (X ∨ Y) ∧ (X ∨ Z)and X ∧ (Y ∨ Z) = (X ∧ Y) ∨ (X ∧ Z) for all X, Y, Z εX, and (2) every element X has a unique complement Xc, defined by X ∨ Xc = Iℒ, X ∧Xc= Oℒ The power lattice P(E) is an atomic complete Boolean lattice, and conversely any atomic complete Boolean lattice has this form.
A Short Tour of Mathematical Morphology on Edge and Vertex Weighted Graphs
Published in Olivier Lézoray, Leo Grady, Image Processing and Analysis with Graphs, 2012
In mathematical morphology, we compare objects with respect to each other. The mathemathical structure that allows us to make such an operation effective is called a lattice. Recall that a (complete) lattice is a partially ordered set, that also has a least upper bound, called supremum, and a greatest lower bound, called infimum. More formally, a lattice [25] (ℒ,≤) is a set ℒ (the space) endowed with an ordering relationship ≤, which is reflexive (∀x∈ℒ,x≤x), anti-symmetric (x ≤ y and y ≤ x ⇒ x = y), and transitive (x ≤ y and y ≤ z ⇒ x ≤ z). This ordering is such that for all x and y, we can define both a larger element x ∨ y and a smaller element x ∧ y. Such a lattice is said to be complete if any subset P of ℒ has a supremum∨P and an infimum∧P that both belongs to ℒ. The supremum is formally the smallest amongst all elements of ℒ that are greater than all the elements of P, and, conversely, the infimum is the largest element of ℒ that is smaller than all the elements of P.
Orness for real m-dimensional interval-valued OWA operators and its application to determine a good partition
Published in International Journal of General Systems, 2019
L. De Miguel, D. Paternain, I. Lizasoain, G. Ochoa, H. Bustince
Throughout this paper denotes a complete lattice, i.e. a partially ordered set in which all subsets have both a supremum and an infimum. and respectively stand for the least and the greatest elements of the lattice L. For more information, see Birkhoff (1967) and Davey and Priestley (1990).