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In mathematics, and more specifically in order theory, several different types of ordered set have been studied. They include: Cyclic orders, orderings in which triples of elements are either clockwise or counterclockwise; Lattices, partial orders in which each pair of elements has a greatest lower bound and a least upper bound.
If is a partial order compatible with the vector space structure of then (,) is called an ordered vector space and is called a vector partial order on . The two axioms imply that translations and positive homotheties are automorphisms of the order structure and the mapping x ↦ − x {\displaystyle x\mapsto -x} is an isomorphism to the dual ...
In crystallography, crystal structure is a description of ordered arrangement of atoms, ions, or molecules in a crystalline material. [1] Ordered structures occur from intrinsic nature of constituent particles to form symmetric patterns that repeat along the principal directions of three-dimensional space in matter.
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).
This is a general situation in order theory: A given order can be inverted by just exchanging its direction, pictorially flipping the Hasse diagram top-down. This yields the so-called dual, inverse, or opposite order. Every order theoretic definition has its dual: it is the notion one obtains by applying the definition to the inverse order.
An order: each number is either less than or greater than any other number. Algebraic structure: there are operations of addition and multiplication, the first of which makes it into a group and the pair of which together make it into a field .
The definition of total order appeared first historically and is a first-order axiomatization of the ordering as a binary predicate. Artin and Schreier gave the definition in terms of positive cone in 1926, which axiomatizes the subcollection of nonnegative elements.
The domain of a structure is an arbitrary set; it is also called the underlying set of the structure, its carrier (especially in universal algebra), its universe (especially in model theory, cf. universe), or its domain of discourse. In classical first-order logic, the definition of a structure prohibits the empty domain. [citation needed] [5]