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.

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.

Lattices as partially ordered sets

If (L, ≤) is a partially ordered set (poset), and SL is an arbitrary subset, then an element uL is said to be an upper bound of S if su for each sS. 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 ux for each upper bound x of S. A set need not have a least upper bound, but it cannot have more than one. Dually, lL is said to be a lower bound of S if ls for each sS. A lower bound l of S is said to be its greatest lower bound, or meet, or infimum, if xl 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 ab and ab 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: a1a2 and b1b2 implies that a1b1a2b2 and a1b1a2b2.

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 1 and a least element 0, which satisfy

0 ≤ x ≤ 1 for every x in L.

The greatest and least element is also called the maximum and minimum, or the top and bottom element, and denoted by ⊤ and ⊥, respectively. Every lattice can be converted 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 ${\displaystyle \bigvee L=a_{1}\lor \cdots \lor a_{n}}$ (resp.${\displaystyle \bigwedge L=a_{1}\land \cdots \land a_{n}}$) where ${\displaystyle L=\{a_{1},\ldots ,a_{n}\}}$.

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 ${\displaystyle \forall a\in \varnothing :x\leq a}$ and ${\displaystyle \forall a\in \varnothing :a\leq x}$, 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 ${\displaystyle \bigvee \varnothing =0}$, and the meet of the empty set is the greatest element ${\displaystyle \bigwedge \varnothing =1}$. 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,

${\displaystyle \bigvee \left(A\cup B\right)=\left(\bigvee A\right)\vee \left(\bigvee B\right)}$

and

${\displaystyle \bigwedge \left(A\cup B\right)=\left(\bigwedge A\right)\wedge \left(\bigwedge B\right)}$

hold. Taking B to be the empty set,

${\displaystyle \bigvee \left(A\cup \emptyset \right)=\left(\bigvee A\right)\vee \left(\bigvee \emptyset \right)=\left(\bigvee A\right)\vee 0=\bigvee A}$

and

${\displaystyle \bigwedge \left(A\cup \emptyset \right)=\left(\bigwedge A\right)\wedge \left(\bigwedge \emptyset \right)=\left(\bigwedge A\right)\wedge 1=\bigwedge A}$

which is consistent with the fact that ${\displaystyle A\cup \emptyset =A}$.

A lattice element y is said to cover another element x, if y > x, but there does not exist a z such that y > z > x. Here, y > x means xy and xy.

A lattice (L, ≤) is called graded, sometimes ranked (but see Ranked poset for an alternative meaning), if it can be equipped with a rank function r from L to ℕ, sometimes to ℤ, compatible with the ordering (so r(x) < r(y) whenever x < y) such that whenever y covers x, then r(y) = r(x) + 1. The value of the rank function for a lattice element is called its rank.

Given a subset of a lattice, HL, meet and join restrict to partial functions – they are undefined if their value is not in the subset H. The resulting structure on H is called a partial lattice. In addition to this extrinsic definition as a subset of some other algebraic structure (a lattice), a partial lattice can also be intrinsically defined as a set with two partial binary operations satisfying certain axioms.[1]