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A lattice in the sense of a 3-dimensional array of regularly spaced points coinciding with e.g. the atom or molecule positions in a crystal, or more generally, the orbit of a group action under translational symmetry, is a translation of the translation lattice: a coset, which need not contain the origin, and therefore need not be a lattice in ...
A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra.It consists of a partially ordered set in which every pair of elements has a unique supremum (also called a least upper bound or join) and a unique infimum (also called a greatest lower bound or meet).
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Every finite poset is directed-complete and algebraic (though not necessarily bounded-complete). Thus any bounded-complete finite poset is a Scott domain. The natural numbers with an additional top element ω constitute an algebraic lattice, hence a Scott domain. For more examples in this direction, see the article on algebraic lattices.
Atomic lattice may refer to: In mineralogy, atomic lattice refers to the arrangement of atoms into a crystal structure. In order theory, a lattice is called an atomic lattice if the underlying partial order is atomic. In chemistry, atomic lattice refers to the arrangement of atoms in an atomic crystalline solid
There is another algebraic lattice that plays an important role in universal algebra: For every algebra A we let Con(A) be the set of all congruence relations on A. Each congruence on A is a subalgebra of the product algebra AxA, so Con(A) ⊆ Sub(AxA). Again we have Con(A), ordered by set inclusion, is a lattice.
In abstract algebra, a residuated lattice is an algebraic structure that is simultaneously a lattice x ≤ y and a monoid x•y which admits operations x\z and z/y, loosely analogous to division or implication, when x•y is viewed as multiplication or conjunction, respectively. Called respectively right and left residuals, these operations ...
A Boolean skew lattice is a symmetric, distributive normal skew lattice with 0, (;,,), such that is a Boolean lattice for each . Given such skew lattice S , a difference operator \ is defined by x \ y = x − x ∧ y ∧ x {\displaystyle x-x\wedge y\wedge x} where the latter is evaluated in the Boolean lattice x ∧ S ∧ x . {\displaystyle x ...