Lattice (order)

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 two elements have a unique supremum (also called a least upper bound or join) and a unique infimum (also called a greatest lower bound or meet). An example is given by the natural numbers, partially ordered by divisibility, for which the unique supremum is the least common multiple and the unique infimum is the greatest common divisor. 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 two elements have a unique supremum (also called a least upper bound or join) and a unique infimum (also called a greatest lower bound or meet). An example is given by the natural numbers, partially ordered by divisibility, for which the unique supremum is the least common multiple and the unique infimum is the greatest common divisor. Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities. Since the two definitions are equivalent, lattice theory draws on both order theory and universal algebra. Semilattices include lattices, which in turn include Heyting and Boolean algebras. These 'lattice-like' structures all admit order-theoretic as well as algebraic descriptions. If (L, ≤) is a partially ordered set (poset), and S ⊆ L is an arbitrary subset, then an element u ∈ L is said to be an upper bound of S if s ≤ u for each s ∈ S. A set may have many upper bounds, or none at all. An upper bound u of S is said to be its least upper bound, or join, or supremum, if u ≤ x for each upper bound x of S. A set need not have a least upper bound, but it cannot have more than one. Dually, l ∈ L is said to be a lower bound of S if l ≤ s for each s ∈ S. A lower bound l of S is said to be its greatest lower bound, or meet, or infimum, if x ≤ l for each lower bound x of S. A set may have many lower bounds, or none at all, but can have at most one greatest lower bound. A partially ordered set (L, ≤) is called a join-semilattice if each two-element subset {a, b} ⊆ L has a join (i.e. least upper bound), and is called a meet-semilattice if each two-element subset has a meet (i.e. greatest lower bound), denoted by a ∨ b and a ∧ b respectively. (L, ≤) is called a lattice if it is both a join- and a meet-semilattice. This definition makes ∨ and ∧ binary operations. Both operations are monotone with respect to the order: a1 ≤ a2 and b1 ≤ b2 implies that a1 ∨ b1 ≤ a2 ∨ b2 and a1 ∧ b1 ≤ a2 ∧ b2. It follows by an induction argument that every non-empty finite subset of a lattice has a least upper bound and a greatest lower bound. With additional assumptions, further conclusions may be possible; see Completeness (order theory) for more discussion of this subject. That article also discusses how one may rephrase the above definition in terms of the existence of suitable Galois connections between related partially ordered sets — an approach of special interest for the category theoretic approach to lattices, and for formal concept analysis. A bounded lattice is a lattice that additionally has a greatest element (also called maximum, or top element, and denoted by 1, or by ⊤ {displaystyle op } ) and a least element (also called minimum, or bottom, denoted by 0 or by ⊥ {displaystyle ot } ), which satisfy Every lattice can be embedded into a bounded lattice by adding an artificial greatest and least element, and every non-empty finite lattice is bounded, by taking the join (resp., meet) of all elements, denoted by ⋁ L = a 1 ∨ ⋯ ∨ a n {displaystyle igvee L=a_{1}lor cdots lor a_{n}} (resp. ⋀ L = a 1 ∧ ⋯ ∧ a n {displaystyle igwedge L=a_{1}land cdots land a_{n}} ) where L = { a 1 , … , a n } {displaystyle L={a_{1},ldots ,a_{n}}} . A partially ordered set is a bounded lattice if and only if every finite set of elements (including the empty set) has a join and a meet. For every element x of a poset it is trivially true (it is a vacuous truth) that ∀ a ∈ ∅ : x ≤ a {displaystyle forall ain varnothing :xleq a} and ∀ a ∈ ∅ : a ≤ x {displaystyle forall ain varnothing :aleq x} , and therefore every element of a poset is both an upper bound and a lower bound of the empty set. This implies that the join of an empty set is the least element ⋁ ∅ = 0 {displaystyle igvee varnothing =0} , and the meet of the empty set is the greatest element ⋀ ∅ = 1 {displaystyle igwedge varnothing =1} . This is consistent with the associativity and commutativity of meet and join: the join of a union of finite sets is equal to the join of the joins of the sets, and dually, the meet of a union of finite sets is equal to the meet of the meets of the sets, i.e., for finite subsets A and B of a poset L,