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{{Short description|Partial order with joins}}
{{stack|{{Binary relations}}}}
In [[Mathematics|mathematics]], a '''join-semilattice''' (or '''upper semilattice''') is a [[Partially ordered set|partially ordered set]] that has a join (a least upper bound) for any nonempty [[Finite set|finite]] [[Subset|subset]]. [[Duality (order theory)|Dually]], a '''meet-semilattice''' (or '''lower semilattice''') is a partially ordered set which has a meet (or greatest lower bound) for any nonempty finite subset. Every join-semilattice is a meet-semilattice in the inverse order and vice versa.

Semilattices can also be defined [[Algebra|algebraically]]: join and meet are associative, commutative, idempotent [[Binary operation|binary operation]]s, 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. 

A [[Lattice (order)|lattice]] is a partially ordered set that is both a meet- and join-semilattice with respect to the same partial order. Algebraically, a lattice is a set with two associative, commutative idempotent binary operations linked by corresponding [[Absorption law|absorption law]]s.

{{Algebraic structures |Lattice}}

==Order-theoretic definition==

A [[Set (mathematics)|set]] {{math|1=''S''}} [[Partially ordered set|partially ordered]] by the [[Binary relation|binary relation]] {{math|1=≤}} is a ''meet-semilattice'' if

: For all elements {{math|1=''x''}} and {{math|1=''y''}} of {{math|1=''S''}}, the greatest lower bound of the set {{math|1={''x'', ''y''} }} exists.

The greatest lower bound of the set {{math|1={''x'', ''y''} }} is called the meet of {{math|1=''x''}} and {{math|1=''y'',}} denoted {{math|1=''x'' ∧ ''y''.}}

Replacing "greatest lower bound" with "least upper bound" results in the dual concept of a ''join-semilattice''. The least upper bound of {{math|1={''x'', ''y''} }} is called the join of {{math|1=''x''}} and {{math|1=''y''}}, denoted {{math|1=''x'' ∨ ''y''}}. Meet and join are [[Binary operation|binary operation]]s on {{math|1=''S''.}} A simple [[Mathematical induction|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).

A join-semilattice is '''bounded''' if it has a least element, the join of the empty set. [[Duality (order theory)|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 (order theory)|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 connection]]s between related posets — an approach of special interest for [[Category theory|category theoretic]] investigations of the concept.

==Algebraic definition==

A ''meet-semilattice'' is an [[Algebraic structure|algebraic structure]] <math>\langle S, \land \rangle</math> consisting of a [[Set (mathematics)|set]] {{math|1=''S''}} with a [[Binary operation|binary operation]] {{math|1=∧}}, called '''meet''', such that for all members {{math|1=''x'', ''y'',}} and {{math|1=''z''}} of {{math|1=''S'',}} the following [[Identity (mathematics)|identities]] hold:

; Associativity: {{math|1=''x'' ∧ (''y'' ∧ ''z'') = (''x'' ∧ ''y'') ∧ ''z''}}
; Commutativity: {{math|1=''x'' ∧ ''y'' = ''y'' ∧ ''x''}}
; Idempotency: {{math|1=''x'' ∧ ''x'' = ''x''}}

A meet-semilattice <math>\langle S, \land \rangle</math> is '''bounded''' if {{math|1=''S''}} includes an [[Identity element|identity element]] 1 such that {{math|1=''x'' ∧ 1 {{=}} ''x''}} for all {{math|1=''x''}} in {{math|1=''S''.}} 

If the symbol {{math|1=∨}}, called '''join''', replaces {{math|1=∧}} in the definition just given, the structure is called a ''join-semilattice''. One can be ambivalent about the particular choice of symbol for the operation, and speak simply of ''semilattices''.

A semilattice is a commutative, idempotent [[Semigroup|semigroup]]; i.e., a commutative [[Band (mathematics)|band]]. A bounded semilattice is an idempotent commutative [[Monoid|monoid]].

A partial order is induced on a meet-semilattice by setting {{math|1=''x'' ≤ ''y''}} whenever {{math|1=''x'' ∧ ''y'' {{=}} ''x''}}. For a join-semilattice, the order is induced by setting {{math|1=''x'' ≤ ''y''}} whenever {{math|1=''x'' ∨ ''y'' {{=}} ''y''}}. In a bounded meet-semilattice, the identity 1 is the greatest element of {{math|1=''S''.}} Similarly, an identity element in a join semilattice is a least element.

== Connection between the two definitions ==
An order theoretic meet-semilattice {{math|1=&lang;''S'', ≤&rang;}} gives rise to a [[Binary operation|binary operation]] {{math|1=∧}} such that {{math|1=&lang;''S'', ∧&rang;}} is an algebraic meet-semilattice. Conversely, the meet-semilattice {{math|1=&lang;''S'', ∧&rang;}} gives rise to a [[Binary relation|binary relation]] {{math|1=≤}} that partially orders {{math|1=''S''}} in the following way: for all elements {{math|1=''x''}} and {{math|1=''y''}} in {{math|1=''S'', ''x'' ≤ ''y''}} if and only if {{math|1=''x'' = ''x'' ∧ ''y''.}}

The relation {{math|1=≤}} introduced in this way defines a partial ordering from which the binary operation {{math|1=∧}} may be recovered. Conversely, the order induced by the algebraically defined semilattice {{math|1=&lang;''S'', ∧&rang;}} coincides with that induced by {{math|1=≤.}}

Hence the two definitions may be used interchangeably, depending on which one is more convenient for a particular purpose. A similar conclusion holds for join-semilattices and the dual ordering ≥.

==Examples==

Semilattices are employed to construct other order structures, or in conjunction with other completeness properties.
* A [[Lattice (order)|lattice]] is both a join- and a meet-semilattice. The interaction of these two semilattices via the [[Absorption law|absorption law]] is what truly distinguishes a lattice from a semilattice.
* The [[Compact element|compact element]]s of an algebraic [[Lattice (order)|lattice]], under the induced partial ordering, form a bounded join-semilattice.
* Any finite semilattice is bounded, by induction.
* A totally ordered set is a [[Distributive lattice|distributive lattice]], hence in particular a meet-semilattice and join-semilattice: any two distinct elements have a greater and lesser one, which are their meet and join.
** A well-ordered set is further a ''bounded'' join-semilattice, as the set as a whole has a least element, hence it is bounded.
*** The [[Natural number#Order|natural numbers]] <math>\mathbb{N}</math>, with their usual order {{math|1=≤,}} are a bounded join-semilattice, with least element 0, although they have no greatest element: they are the smallest infinite well-ordered set.
* Any single-rooted [[Tree (set theory)|tree]] (with the single root as the least element) of height <math>\leq \omega</math> is a (generally unbounded) meet-semilattice. Consider for example the set of finite words over some alphabet, ordered by the [[Prefix order|prefix order]]. It has a least element (the empty word), which is an annihilator element of the meet operation, but no greatest (identity) element.
* A [[Scott domain]] is a meet-semilattice.
* Membership in any set {{math|1=''L''}} can  be taken as a [[Model theory|model]] of a semilattice with base set {{math|1=''L'',}} because a semilattice captures the essence of set [[Extensionality|extensionality]]. Let {{math|1=''a'' ∧ ''b''}} denote {{math|1=''a'' ∈ ''L''}} & {{math|1=''b'' ∈ ''L''.}} Two sets differing only in one or both of the:
# Order in which their members are listed;
# Multiplicity of one or more members,
:are in fact the same set. Commutativity and associativity of {{math|1=∧}} assure (1), [[Idempotence|idempotence]], (2). This semilattice is the free semilattice over {{math|1=''L''.}}  It is not bounded by {{math|1=''L'',}} because a set is not a member of itself.
* Classical extensional [[Mereology|mereology]] defines a join-semilattice, with join read as binary fusion. This semilattice is bounded from above by the world individual.
* Given a set {{math|1=''S'',}} the collection of partitions <math> \xi </math> of {{math|1=''S''}} is a join-semilattice. In fact, the partial order is given by <math> \xi \leq \eta </math> if <math> \forall Q \in \eta, \exists P \in \xi </math> such that <math> Q \subset P </math> and the join of two partitions is given by <math> \xi \vee \eta = \{ P \cap Q \mid P \in \xi \ \& \ Q \in \eta \} </math>.  This semilattice is bounded, with the least element being the singleton partition <math> \{ S \} </math>.

==Semilattice morphisms==

The above algebraic definition of a semilattice suggests a notion of [[Morphism|morphism]] between two semilattices. Given two join-semilattices {{math|1=(''S'', ∨)}} and {{math|1=(''T'', ∨)}}, a [[Homomorphism|homomorphism]] of (join-) semilattices is a function {{math|1=''f'': ''S'' → ''T''}} such that

:{{math|1=''f''(''x'' ∨ ''y'') = ''f''(''x'') ∨ ''f''(''y'').}}

Hence {{math|1=''f''}} is just a homomorphism of the two semigroups associated with each semilattice. If {{math|1=''S''}} and {{math|1=''T''}} both include a least element 0, then {{math|1=''f''}} should also be a [[Monoid|monoid]] homomorphism, i.e. we additionally require that

: {{math|1=''f''(0) = 0.}}

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 {{math|1=∧}} with {{math|1=∨}} 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. For an explanation see the entry preservation of limits.

==Equivalence with algebraic lattices==

There is a well-known [[Equivalence of categories|equivalence]] between the category <math>\mathcal{S}</math> of join-semilattices with zero with <math>(\vee,0)</math>-homomorphisms and the category <math>\mathcal{A}</math> of algebraic lattices with [[Compact element|compactness]]-preserving complete join-homomorphisms, as follows. With a join-semilattice <math>S</math> with zero, we associate its ideal lattice <math>\operatorname{Id}\ S</math>. With a <math>(\vee,0)</math>-homomorphism <math>f \colon S \to T</math> of <math>(\vee,0)</math>-semilattices, we associate the map <math>\operatorname{Id}\ f \colon \operatorname{Id}\ S \to \operatorname{Id}\ T</math>, that with any ideal <math>I</math> of <math>S</math> associates the ideal of <math>T</math> generated by <math>f(I)</math>. This defines a functor <math>\operatorname{Id} \colon \mathcal{S} \to \mathcal{A}</math>. Conversely, with every algebraic lattice <math>A</math> we associate the <math>(\vee,0)</math>-semilattice <math>K(A)</math> of all [[Compact element|compact element]]s of <math>A</math>, and with every compactness-preserving complete join-homomorphism <math>f \colon A \to B</math> between algebraic lattices we associate the restriction <math>K(f) \colon K(A) \to K(B)</math>. This defines a functor <math>K \colon \mathcal{A} \to \mathcal{S}</math>. The pair <math>(\operatorname{Id},K)</math> defines a category equivalence between <math>\mathcal{S}</math> and <math>\mathcal{A}</math>.

==Distributive semilattices==

Surprisingly, there is a notion of "distributivity" applicable to semilattices, even though distributivity conventionally requires the interaction of two binary operations. This notion requires but a single operation, and generalizes the distributivity condition for lattices. A join-semilattice is '''distributive''' if for all {{math|1=''a'', ''b'',}} and {{math|1=''x''}} with {{math|1=''x'' &le; ''a'' &or; ''b''}} there exist {{math|1=''a' '' &le; ''a''}} and {{math|1=''b' '' &le; ''b''}} such that {{math|1=''x'' = ''a' '' &or; ''b' ''.}} Distributive meet-semilattices are defined dually. These definitions are justified by the fact that any distributive join-semilattice in which binary meets exist is a distributive lattice. See the entry [[Distributivity (order theory)|distributivity (order theory)]].

A join-semilattice is distributive if and only if the lattice of its [[Ideal (order theory)|ideals]] (under inclusion) is distributive.

==Complete semilattices==

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 lattice|complete lattice]]s. 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)|completeness (order theory)]].

Nevertheless, the literature on occasion still takes complete join- or meet-semilattices to be complete lattices. In this case, "completeness" denotes a restriction on the scope of the [[Homomorphism|homomorphism]]s. 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.

Another usage of "complete meet-semilattice" refers to a bounded complete [[Complete partial order|cpo]]. A complete meet-semilattice in this sense is arguably the "most complete" meet-semilattice that is not necessarily a complete lattice. Indeed, a complete meet-semilattice has all ''non-empty'' meets (which is equivalent to being bounded complete) and all [[Directed set|directed]] joins. If such a structure has also a greatest element (the meet of the empty set), it is also a complete lattice. Thus a complete semilattice turns out to be "a complete lattice possibly lacking a top". This definition is of interest specifically in [[Domain theory|domain theory]], where bounded complete algebraic cpos are studied as [[Scott domain]]s. Hence Scott domains have been called ''algebraic semilattices''.

Cardinality-restricted notions of completeness for semilattices have been rarely considered in the literature.<ref>E. G. Manes, ''Algebraic theories'', Graduate Texts in Mathematics Volume 26, Springer 1976, p. 57</ref>

==Free semilattices==

This section presupposes some knowledge of [[Category theory|category theory]]. In various situations, [[Free object|free]] semilattices exist. For example, the [[Forgetful functor|forgetful functor]] from the category of join-semilattices (and their homomorphisms) to the [[Category theory|category]] of sets (and functions) admits a [[Adjoint functors|left adjoint]]. Therefore, the free join-semilattice {{math|1='''F'''(''S'')}} over a set {{math|1=''S''}} is constructed by taking the collection of all non-empty ''finite'' [[Subset|subset]]s of {{math|1=''S'',}} ordered by subset inclusion. Clearly, {{math|1=''S''}} can be embedded into {{math|1='''F'''(''S'')}} by a mapping {{math|1=''e''}} that takes any element {{math|1=''s''}} in {{math|1=''S''}} to the singleton set {{math|1={''s''}.}} Then any function {{math|1=''f''}} from a {{math|1=''S''}} to a join-semilattice {{math|1=''T''}} (more formally, to the underlying set of {{math|1=''T''}}) induces a unique homomorphism {{math|1=''f' ''}} between the join-semilattices {{math|1='''F'''(''S'')}} and {{math|1=''T'',}} such that {{math|1=''f'' = ''f' '' ○ ''e''.}} Explicitly, {{math|1=''f' ''}} is given by <math display=inline>f'(A) = \bigvee\{f(s) | s \in A\}.</math> Now the obvious uniqueness of {{math|1=''f' ''}} suffices to obtain the required adjunction—the morphism-part of the functor {{math|1='''F'''}} can be derived from general considerations (see [[Adjoint functors|adjoint functors]]). 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. Notably, both the forgetful functors from the category of [[Complete Heyting algebra|frames]] and frame-homomorphisms, and from the category of distributive lattices and lattice-homomorphisms, have a left adjoint.

==See also==

* {{annotated link|Directed set}} − generalization of join semilattice
* {{annotated link|Semiring}}

==Notes==

{{reflist}}

==References==

* {{Cite book |last1=Davey |first1=B. A. |last2=Priestley |first2=H. A. |title=Introduction to Lattices and Order|title-link= Introduction to Lattices and Order |publisher={{wipe|Cambridge University Press}} |edition=second |year=2002 |isbn=0-521-78451-4}}
* {{Cite book |last=Vickers |first=Steven |title=Topology via Logic |publisher={{wipe|Cambridge University Press}} |year=1989 |isbn=0-521-36062-5}}

It is often the case that standard treatments of lattice theory define a semilattice, if that, and then say no more. See the references in the entries [[Order theory|order theory]] and lattice theory. Moreover, there is no literature on semilattices of comparable magnitude to that on [[Semigroup|semigroup]]s.

==External links==

* Jipsen's algebra structures page: [http://math.chapman.edu/~jipsen/structures/doku.php/semilattices Semilattices.]


[[Category:Lattice theory]]
[[Category:Algebraic structures]]

{{Sourceattribution|Semilattice}}