Y indicates that the column's property is always true the row's term (at the very left), while ✗ indicates that the property is not guaranteed in general (it might, or might not, hold). For example, that every equivalence relation is symmetric, but not necessarily antisymmetric, is indicated by Y in the "Symmetric" column and ✗ in the "Antisymmetric" column, respectively.
The sub-field of
abstract algebra that studies lattices is called lattice theory.
A lattice can be defined either order-theoretically as a partially ordered set, or as an algebraic structure.
As partially ordered set
partially ordered set (poset) is called a lattice if it is both a join- and a meet-
semilattice, i.e. each two-element subset has a
join (i.e. least upper bound, denoted by ) and
meet (i.e. greatest lower bound, denoted by ). This definition makes and binary operations. Both operations are monotone with respect to the given order: and implies that and
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; seeCompleteness (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.
Given a subset of a lattice, meet and join restrict to
partial functions – they are undefined if their value is not in the subset The resulting structure on 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.
As algebraic structure
A lattice is an
algebraic structure, consisting of a set and two binary, commutative and associative
operations and on satisfying the following axiomatic identities for all elements (sometimes called absorption laws):
The following two identities are also usually regarded as axioms, even though they follow from the two absorption laws taken together. These are called idempotent laws.
These axioms assert that both and are
semilattices. The absorption laws, the only axioms above in which both meet and join appear, distinguish a lattice from an arbitrary pair of semilattice structures and assure that the two semilattices interact appropriately. In particular, each semilattice is the
dual of the other. The absorption laws can be viewed as a requirement that the meet and join semilattices define the same
Connection between the two definitions
An order-theoretic lattice gives rise to the two binary operations and Since the commutative, associative and absorption laws can easily be verified for these operations, they make into a lattice in the algebraic sense.
The converse is also true. Given an algebraically defined lattice one can define a partial order on by setting
for all elements The laws of absorption ensure that both definitions are equivalent:
and dually for the other direction.
One can now check that the relation ≤ introduced in this way defines a partial ordering within which binary meets and joins are given through the original operations and
Since the two definitions of a lattice are equivalent, one may freely invoke aspects of either definition in any way that suits the purpose at hand.
A bounded lattice is a lattice that additionally has a greatest element (also called maximum, or top element, and denoted by 1, or by ) and a least element (also called minimum, or bottom, denoted by 0 or by ), which satisfy
A bounded lattice may also be defined as an algebraic structure of the form such that is a lattice, (the lattice's bottom) is the
identity element for the join operation and (the lattice's top) is the identity element for the meet operation
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 of a poset it is
vacuously true that
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 and the meet of the empty set is the greatest element 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, that is, for finite subsets of a poset
hold. Taking B to be the empty set,
which is consistent with the fact that
Every lattice can be embedded into a bounded lattice by adding a greatest and a least element. Furthermore, every non-empty finite lattice is bounded, by taking the join (respectively, meet) of all elements, denoted by (respectively ) where is the set of all elements.
Connection to other algebraic structures
Lattices have some connections to the family of
group-like algebraic structures. Because meet and join both commute and associate, a lattice can be viewed as consisting of two commutative
semigroups having the same domain. For a bounded lattice, these semigroups are in fact commutative
absorption law is the only defining identity that is peculiar to lattice theory.
By commutativity, associativity and idempotence one can think of join and meet as operations on non-empty finite sets, rather than on pairs of elements. In a bounded lattice the join and meet of the empty set can also be defined (as and respectively). This makes bounded lattices somewhat more natural than general lattices, and many authors require all lattices to be bounded.
The algebraic interpretation of lattices plays an essential role in
Pic. 2: Lattice of integer divisors of 60, ordered by "divides".
Pic. 3: Lattice of
partitions of ordered by "refines".
Pic. 4: Lattice of positive integers, ordered by
Pic. 5: Lattice of nonnegative integer pairs, ordered componentwise.
For any set the collection of all subsets of (called the
power set of ) can be ordered via
subset inclusion to obtain a lattice bounded by itself and the empty set. In this lattice, the supremum is provided by
set union and the infimum is provided by
set intersection (see Pic. 1).
For any set the collection of all finite subsets of ordered by inclusion, is also a lattice, and will be bounded if and only if is finite.
The set of
compact elements of an
arithmetic complete lattice is a lattice with a least element, where the lattice operations are given by restricting the respective operations of the arithmetic lattice. This is the specific property that distinguishes arithmetic lattices from
algebraic lattices, for which the compacts only form a
join-semilattice. Both of these classes of complete lattices are studied in
Further examples of lattices are given for each of the additional properties discussed below.
Examples of non-lattices
Pic. 8: Non-lattice poset: and have common lower bounds and but none of them is the
greatest lower bound.
Pic. 7: Non-lattice poset: and have common upper bounds and but none of them is the
least upper bound.
Pic. 6: Non-lattice poset: and have no common upper bound.
Most partially ordered sets are not lattices, including the following.
A discrete poset, meaning a poset such that implies is a lattice if and only if it has at most one element. In particular the two-element discrete poset is not a lattice.
Although the set partially ordered by divisibility is a lattice, the set so ordered is not a lattice because the pair 2, 3 lacks a join; similarly, 2, 3 lacks a meet in
The set partially ordered by divisibility is not a lattice. Every pair of elements has an upper bound and a lower bound, but the pair 2, 3 has three upper bounds, namely 12, 18, and 36, none of which is the least of those three under divisibility (12 and 18 do not divide each other). Likewise the pair 12, 18 has three lower bounds, namely 1, 2, and 3, none of which is the greatest of those three under divisibility (2 and 3 do not divide each other).
Morphisms of lattices
Pic. 9: Monotonic map between lattices that preserves neither joins nor meets, since and
The appropriate notion of a
morphism between two lattices flows easily from the
above algebraic definition. Given two lattices and a lattice homomorphism from L to M is a function such that for all
Thus is a
homomorphism of the two underlying
semilattices. When lattices with more structure are considered, the morphisms should "respect" the extra structure, too. In particular, a bounded-lattice homomorphism (usually called just "lattice homomorphism") between two bounded lattices and should also have the following property:
In the order-theoretic formulation, these conditions just state that a homomorphism of lattices is a function
preserving binary meets and joins. For bounded lattices, preservation of least and greatest elements is just preservation of join and meet of the empty set.
Any homomorphism of lattices is necessarily
monotone with respect to the associated ordering relation; see
Limit preserving function. The converse is not true: monotonicity by no means implies the required preservation of meets and joins (see Pic. 9), although an
order-preservingbijection is a homomorphism if its
inverse is also order-preserving.
Given the standard definition of
isomorphisms as invertible morphisms, a lattice isomorphism is just a
bijective lattice homomorphism. Similarly, a lattice endomorphism is a lattice homomorphism from a lattice to itself, and a lattice automorphism is a bijective lattice endomorphism. Lattices and their homomorphisms form a
Let and be two lattices with 0 and 1. A homomorphism from to is called 0,1-separatingif and only if ( separates 0) and ( separates 1).
A sublattice of a lattice is a subset of that is a lattice with the same meet and join operations as That is, if is a lattice and is a subset of such that for every pair of elements both and are in then is a sublattice of 
A sublattice of a lattice is a convex sublattice of if and implies that belongs to for all elements
A poset is called a complete lattice if all its subsets have both a join and a meet. In particular, every complete lattice is a bounded lattice. While bounded lattice homomorphisms in general preserve only finite joins and meets, complete lattice homomorphisms are required to preserve arbitrary joins and meets.
Every poset that is a complete semilattice is also a complete lattice. Related to this result is the interesting phenomenon that there are various competing notions of homomorphism for this class of posets, depending on whether they are seen as complete lattices, complete join-semilattices, complete meet-semilattices, or as join-complete or meet-complete lattices.
Note that "partial lattice" is not the opposite of "complete lattice" – rather, "partial lattice", "lattice", and "complete lattice" are increasingly restrictive definitions.
A conditionally complete lattice is a lattice in which every nonempty subset that has an upper bound has a join (that is, a least upper bound). Such lattices provide the most direct generalization of the
completeness axiom of the
real numbers. A conditionally complete lattice is either a complete lattice, or a complete lattice without its maximum element its minimum element or both.
Pic. 11: Smallest non-modular (and hence non-distributive) lattice N5. The labelled elements violate the distributivity equation but satisfy its dual
Since lattices come with two binary operations, it is natural to ask whether one of them
distributes over the other, that is, whether one or the other of the following
dual laws holds for every three elements :
Distributivity of over
Distributivity of over
A lattice that satisfies the first or, equivalently (as it turns out), the second axiom, is called a distributive lattice.
The only non-distributive lattices with fewer than 6 elements are called M3 and N5; they are shown in Pictures 10 and 11, respectively. A lattice is distributive if and only if it doesn't have a
sublattice isomorphic to M3 or N5. Each distributive lattice is isomorphic to a lattice of sets (with union and intersection as join and meet, respectively).
For some applications the distributivity condition is too strong, and the following weaker property is often useful. A lattice is modular if, for all elements the following identity holds:
This condition is equivalent to the following axiom:
implies (Modular law)
A lattice is modular if and only if it doesn't have a
sublattice isomorphic to N5 (shown in Pic. 11).
Besides distributive lattices, examples of modular lattices are the lattice of
two-sided ideals of a
ring, the lattice of submodules of a
module, and the lattice of
normal subgroups of a
set of first-order terms with the ordering "is more specific than" is a non-modular lattice used in
A finite lattice is modular if and only if it is both upper and lower
semimodular. For a graded lattice, (upper) semimodularity is equivalent to the following condition on the rank function
Another equivalent (for graded lattices) condition is
for each and in if and both cover then covers both and
A lattice is called lower semimodular if its dual is semimodular. For finite lattices this means that the previous conditions hold with and exchanged, "covers" exchanged with "is covered by", and inequalities reversed.
Continuity and algebraicity
domain theory, it is natural to seek to approximate the elements in a partial order by "much simpler" elements. This leads to the class of
continuous posets, consisting of posets where every element can be obtained as the supremum of a
directed set of elements that are
way-below the element. If one can additionally restrict these to the
compact elements of a poset for obtaining these directed sets, then the poset is even
algebraic. Both concepts can be applied to lattices as follows:
Both of these classes have interesting properties. For example, continuous lattices can be characterized as algebraic structures (with infinitary operations) satisfying certain identities. While such a characterization is not known for algebraic lattices, they can be described "syntactically" via
Scott information systems.
Let be a bounded lattice with greatest element 1 and least element 0. Two elements and of are complements of each other if and only if:
In general, some elements of a bounded lattice might not have a complement, and others might have more than one complement. For example, the set with its usual ordering is a bounded lattice, and does not have a complement. In the bounded lattice N5, the element has two complements, viz. and (see Pic. 11). A bounded lattice for which every element has a complement is called a
A complemented lattice that is also distributive is a
Boolean algebra. For a distributive lattice, the complement of when it exists, is unique.
In the case the complement is unique, we write ¬x = y and equivalently, ¬y = x. The corresponding unary
operation over called complementation, introduces an analogue of logical
negation into lattice theory.
Heyting algebras are an example of distributive lattices where some members might be lacking complements. Every element of a Heyting algebra has, on the other hand, a
pseudo-complement, also denoted ¬x. The pseudo-complement is the greatest element such that If the pseudo-complement of every element of a Heyting algebra is in fact a complement, then the Heyting algebra is in fact a Boolean algebra.
Jordan–Dedekind chain condition
A chain from to is a set where
The length of this chain is n, or one less than its number of elements. A chain is maximal if covers for all
If for any pair, and where all maximal chains from to have the same length, then the lattice is said to satisfy the Jordan–Dedekind chain condition.
A lattice is called graded, sometimes ranked (but see
Ranked poset for an alternative meaning), if it can be equipped with a rank function sometimes to ℤ, compatible with the ordering (so whenever ) such that whenever covers then The value of the rank function for a lattice element is called its rank.
A lattice element is said to
cover another element if but there does not exist a such that
Here, means and
Any set may be used to generate the free semilattice The free semilattice is defined to consist of all of the finite subsets of with the semilattice operation given by ordinary
set union. The free semilattice has the
universal property. For the free lattice over a set Whitman gave a construction based on polynomials over 's members.
Important lattice-theoretic notions
We now define some order-theoretic notions of importance to lattice theory. In the following, let be an element of some lattice If has a bottom element is sometimes required. is called:
Join irreducible if implies for all When the first condition is generalized to arbitrary joins is called completely join irreducible (or -irreducible). The dual notion is meet irreducibility (-irreducible). For example, in Pic. 2, the elements 2, 3, 4, and 5 are join irreducible, while 12, 15, 20, and 30 are meet irreducible. In the lattice of
real numbers with the usual order, each element is join irreducible, but none is completely join irreducible.
Join prime if implies This too can be generalized to obtain the notion completely join prime. The dual notion is meet prime. Every join-prime element is also join irreducible, and every meet-prime element is also meet irreducible. The converse holds if is distributive.
Let have a bottom element 0. An element of is an
atom if and there exists no element such that Then is called:
Atomic if for every nonzero element of there exists an atom of such that 
However, many sources and mathematical communities use the term "atomic" to mean "atomistic" as defined above.
The notions of
ideals and the dual notion of
filters refer to particular kinds of
subsets of a partially ordered set, and are therefore important for lattice theory. Details can be found in the respective entries.
Join and meet – two related operations on a poset in order theory