“lattice”的概念、定义、翻译、参考文献-科学参考

lattice

Physics

The regular arrangement of atoms, ions, or molecules in a crystalline solid. See crystal lattice.

Mathematics

A partially ordered set X in which every pair of elements x,y has a least upper bound x ∨ y and a greatest lower bound x ∧ y. The lattice is complete if every subset of X has a least upper bound and greatest lower bound. A power set with ∨ = ∪ and ∧ = ∩ is a complete lattice. ℕ with ≤ = | (divides), ∨ = lcm, and ∧ = hcf is an incomplete lattice as the elements of ℕ have no upper bound.
Linearly independent vectors v₁,…, v_k in ℝⁿ generate a lattice of vectors a₁v₁ + ∙∙∙ + a_kv_k where a₁,…, a_k are integers. ℤⁿ in ℝⁿ is often referred to as the integer lattice or cubic lattice; it is the lattice generated by the canonical basis.

Chemistry

The regular arrangement of atoms, ions, or molecules in a crystalline solid. See crystal lattice.

Chemical Engineering

The regular and organized arrangement of atoms, ions, or molecules in a stationary structure, such as in a crystal. The arrangement is usually three-dimensional such as in a diamond. However, two-dimensional crystal structures exist such as graphite which has a layer lattice in which the atoms of carbon are chemically bonded in planes, with relatively weak forces between the planes. The lattice energy is the energy that is released when one mole of a solid ionic compound is formed from its constituent gaseous ions. It is alternatively defined as the energy required to separate completely the constituent ions of a crystal from each other to an infinite distance. The lattice energy is effectively a measure of the stability of a crystal lattice.

Computer

An algebraic structure, such as a Boolean algebra, in which there are two dyadic operations that are both commutative and associative and satisfy the absorption and idempotent laws. The two dyadic operators, denoted by ∧ and ∨, are called the meet and the join respectively.
An alternative but equivalent view of a lattice is as a set L on which there is a partial ordering defined. Further, every pair of elements has both a greatest lower bound and a least upper bound. The least upper bound of {x,y} can be denoted by x ∨ y and is referred to as the join of x and y. The greatest lower bound can be denoted by x ∧ y and is called the meet of x and y. It can then be shown that these operations satisfy the properties mentioned in the earlier definition, since a partial ordering ≤ can be introduced by defining
$a \leq b iff a \lor b = a$
Lattices in the form of Boolean algebras play a very important role in much of the theory and mathematical ideas underlying computer science. Lattices are also basic to much of the approximation theory underlying the ideas of denotational semantics.

Electronics and Electrical Engineering

See crystal lattice.

Geology and Earth Sciences

A regular, three-dimensional framework which indicates the ordered arrangement of atoms in crystals. The smallest complete lattice is known as the ‘unit cell’ and it may be repeated many times to form a complete crystal. The shape of the unit cell varies according to the arrangement of the points of the lattice in space, i.e. a ‘space lattice’.

单词	lattice
释义	lattice Physics The regular arrangement of atoms, ions, or molecules in a crystalline solid. See crystal lattice. Mathematics A partially ordered set X in which every pair of elements x,y has a least upper bound x ∨ y and a greatest lower bound x ∧ y. The lattice is complete if every subset of X has a least upper bound and greatest lower bound. A power set with ∨ = ∪ and ∧ = ∩ is a complete lattice. ℕ with ≤ = \| (divides), ∨ = lcm, and ∧ = hcf is an incomplete lattice as the elements of ℕ have no upper bound. Linearly independent vectors v₁,…, v_k in ℝⁿ generate a lattice of vectors a₁v₁ + ∙∙∙ + a_kv_k where a₁,…, a_k are integers. ℤⁿ in ℝⁿ is often referred to as the integer lattice or cubic lattice; it is the lattice generated by the canonical basis. Chemistry The regular arrangement of atoms, ions, or molecules in a crystalline solid. See crystal lattice. Chemical Engineering The regular and organized arrangement of atoms, ions, or molecules in a stationary structure, such as in a crystal. The arrangement is usually three-dimensional such as in a diamond. However, two-dimensional crystal structures exist such as graphite which has a layer lattice in which the atoms of carbon are chemically bonded in planes, with relatively weak forces between the planes. The lattice energy is the energy that is released when one mole of a solid ionic compound is formed from its constituent gaseous ions. It is alternatively defined as the energy required to separate completely the constituent ions of a crystal from each other to an infinite distance. The lattice energy is effectively a measure of the stability of a crystal lattice. Computer An algebraic structure, such as a Boolean algebra, in which there are two dyadic operations that are both commutative and associative and satisfy the absorption and idempotent laws. The two dyadic operators, denoted by ∧ and ∨, are called the meet and the join respectively. An alternative but equivalent view of a lattice is as a set L on which there is a partial ordering defined. Further, every pair of elements has both a greatest lower bound and a least upper bound. The least upper bound of {x,y} can be denoted by x ∨ y and is referred to as the join of x and y. The greatest lower bound can be denoted by x ∧ y and is called the meet of x and y. It can then be shown that these operations satisfy the properties mentioned in the earlier definition, since a partial ordering ≤ can be introduced by defining $a \leq b iff a \lor b = a$ Lattices in the form of Boolean algebras play a very important role in much of the theory and mathematical ideas underlying computer science. Lattices are also basic to much of the approximation theory underlying the ideas of denotational semantics. Electronics and Electrical Engineering See crystal lattice. Geology and Earth Sciences A regular, three-dimensional framework which indicates the ordered arrangement of atoms in crystals. The smallest complete lattice is known as the ‘unit cell’ and it may be repeated many times to form a complete crystal. The shape of the unit cell varies according to the arrangement of the points of the lattice in space, i.e. a ‘space lattice’.
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