A term's definition may require additional properties that are not listed in this table.
Semilattices can also be defined algebraically: join and meet are associative, commutative, idempotent binary operations, and any such operation induces a partial order (and the respective inverse order) such that the result of the operation for any two elements is the least upper bound (or greatest lower bound) of the elements with respect to this partial order.
Algebraically, a lattice is a set with two associative, commutative idempotent binary operations linked by corresponding absorption laws.
Meet and join are binary operations on S. A simple induction argument shows that the existence of all possible pairwise suprema (infima), as per the definition, implies the existence of all non-empty finite suprema (infima).
Dually, a meet-semilattice is bounded if it has a greatest element, the meet of the empty set.
Other properties may be assumed; see the article on completeness in order theory for more discussion on this subject.
That article also discusses how we may rephrase the above definition in terms of the existence of suitable Galois connections between related posets — an approach of special interest for category theoretic investigations of the concept.
consisting of a set S with a binary operation ∧, called meet, such that for all members x, y, and z of S, the following identities hold: A meet-semilattice
One can be ambivalent about the particular choice of symbol for the operation, and speak simply of semilattices.
Conversely, the meet-semilattice ⟨S, ∧⟩ gives rise to a binary relation ≤ that partially orders S in the following way: for all elements x and y in S, x ≤ y if and only if x = x ∧ y.
The relation ≤ introduced in this way defines a partial ordering from which the binary operation ∧ may be recovered.
Semilattices are employed to construct other order structures, or in conjunction with other completeness properties.
If S and T both include a least element 0, then f should also be a monoid homomorphism, i.e. we additionally require that In the order-theoretic formulation, these conditions just state that a homomorphism of join-semilattices is a function that preserves binary joins and least elements, if such there be.
The obvious dual—replacing ∧ with ∨ and 0 with 1—transforms this definition of a join-semilattice homomorphism into its meet-semilattice equivalent.
Note that any semilattice homomorphism is necessarily monotone with respect to the associated ordering relation.
This notion requires but a single operation, and generalizes the distributivity condition for lattices.
Nowadays, the term "complete semilattice" has no generally accepted meaning, and various mutually inconsistent definitions exist.
If completeness is taken to require the existence of all infinite joins, or all infinite meets, whichever the case may be, as well as finite ones, this immediately leads to partial orders that are in fact complete lattices.
For why the existence of all possible infinite joins entails the existence of all possible infinite meets (and vice versa), see the entry completeness (order theory).
In this case, "completeness" denotes a restriction on the scope of the homomorphisms.
Specifically, a complete join-semilattice requires that the homomorphisms preserve all joins, but contrary to the situation we find for completeness properties, this does not require that homomorphisms preserve all meets.
On the other hand, we can conclude that every such mapping is the lower adjoint of some Galois connection.
The corresponding (unique) upper adjoint will then be a homomorphism of complete meet-semilattices.
This gives rise to a number of useful categorical dualities between the categories of all complete semilattices with morphisms preserving all meets or joins, respectively.
If such a structure has also a greatest element (the meet of the empty set), it is also a complete lattice.
Cardinality-restricted notions of completeness for semilattices have been rarely considered in the literature.
The case of free meet-semilattices is dual, using the opposite subset inclusion as an ordering.
For join-semilattices with bottom, we just add the empty set to the above collection of subsets.
In addition, semilattices often serve as generators for free objects within other categories.
It is often the case that standard treatments of lattice theory define a semilattice, if that, and then say no more.