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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).
A partially ordered set that is both a join-semilattice and a meet-semilattice is a lattice. A lattice in which every subset, not just every pair, possesses a meet and a join is a complete lattice. It is also possible to define a partial lattice, in which not all pairs have a meet or join but the operations (when defined) satisfy certain axioms ...
It is also algebraic, since every finite word happens to be compact and we certainly can approximate infinite words by chains of finite ones. Thus this is a Scott domain which is not an algebraic lattice. For a negative example, consider the real numbers in the unit interval [0,1], ordered by their natural order. This bounded-complete dcpo is ...
Addition and multiplication are prototypical examples of operations that combine two elements of a set to produce a third element of the same set. These operations obey several algebraic laws. For example, a + (b + c) = (a + b) + c and a(bc) = (ab)c are associative laws, and a + b = b + a and ab = ba are commutative laws. Many systems studied ...
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
In mathematics, many types of algebraic structures are studied. Abstract algebra is primarily the study of specific algebraic structures and their properties. Algebraic structures may be viewed in different ways, however the common starting point of algebra texts is that an algebraic object incorporates one or more sets with one or more binary operations or unary operations satisfying a ...
Consider a partially ordered set (P, ≤) that is a complete lattice.Then P is a complete Heyting algebra or frame if any of the following equivalent conditions hold: . P is a Heyting algebra, i.e. the operation () has a right adjoint (also called the lower adjoint of a (monotone) Galois connection), for each element x of P.
Cheung (1974) defines the adjoint of a geometric lattice (or of the matroid defined from it) to be a minimal geometric lattice into which the dual lattice of is order-embedded. Some matroids do not have adjoints; an example is the Vámos matroid. [6]
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