An Introduction to the Theory of Lattice Ý Jinfang Wang £

Home , Algebraic structure, Join and meet, Lattice (order), Semilattice

Graduate School of Science and Technology, Chiba University 1-33 Yayoi-cho, Inage-ku, Chiba 263-8522, Japan

May 11, 2006

£ Fax: 81-43-290-3663. E-Mail addresses: [email protected]

Ý Partially supported by Grand-in-Aid 15500179 of the Japanese Ministry of Education, Science, Sports and Cul- ture.

1 1 Introduction

A lattice1 is a partially ordered set (or poset), in which all nonempty ﬁnite subsets have both a supremum (join) and an inﬁmum (meet). Lattices can also be characterized as algebraic structures that satisfy certain identities. Since both views can be used interchangeably, lattice theory can draw upon applications and methods both from order theory and from universal algebra. Lattices constitute one of the most prominent representatives of a series of “lattice-like” structures which admit order-theoretic as well as algebraic descriptions, such as semilattices, Heyting algebras, and Boolean algebras.

2 Semilattice

A semilattice is a partially ordered set within which either all binary sets have a supremum (join) or all binary sets have an inﬁmum (meet). Consequently, one speaks of either a join-semilattice or a meet-semilattice. Semilattices may be regarded as a generalization of the more prominent concept of a lattice. In the literature, join-semilattices sometimes are sometimes additionally required to have a least element (the join of the empty set). Dually, meet-semilattices may include a greatest element.

2.1 Deﬁnitions

Semilattices as posets

Ë µ Ü

DEFINITION 2.1 (MEET-SEMILATTICE). A poset ´ is a meet-semilattice if for all elements

Ë Ü Ý

and Ý of , the greatest lower bound (meet) of the set exists.

Ë µ Ü Ý Ë

Dually, we deﬁne a poset ´ as a join-semilattice, if for all elements and of , the least

Ü Ý

upper bound (join) of the set exists.

Ý Ü Ý Ü Ý The join and meet of Ü and are denoted by and , respectively. Clearly, and deﬁne binary operations on semilattices. By induction, it is easy to see that the existence of binary

suprema implies the existence of all non-empty ﬁnite suprema. The same holds true for inﬁma.

Ë µ Semilattices as algebraic structures. Consider an algebraic structure given by ´ ,

being a binary operation.

Ë µ

DEFINITION 2.2 (MEET-SEMILATTICE). A poset ´ is a meet-semilattice if the following iden-

Ý Þ Ë

tities hold for all elements Ü, , and in :

´Ý Þ µ´Ü Ý µ Þ

Ü (associativity)

Ý Ý Ü

Ü (commutativity)

Ü Ü Ü (idempotency)

1The term ”lattice” derives from the shape of the Hasse diagrams that result from depicting the orders.

´Ë µ In this case, is called meet. A join-semilattice is an algebraic structure , where the

meet satisﬁes the same axioms as above, the difference in notation being only for convenience,

usually for dual interpretations in speciﬁc applications.

Ë µ ´Ë µ Deﬁnition 2.2 says that ´ is a semilattice if is an idempotent, commutative semi-

group. Sometimes it is desirable to consider meet-, join-semilattices with greatest (least) elements,

Ë ½µ

which are deﬁned as idempotent, commutative monoids. Explicitly, ´ are meet-semilattices

Ë µ

with greatestelements if ´ are meet-semilattices and in addition we have

Ü ½Ü

¾ Ë ´Ë ¼µ ´Ë µ for all Ü . Similarly, are join-semilattices with least elements if are join-

semilattices and

Ü ¼Ü

¾ Ë for all Ü .

Equivalence of both deﬁnitions. An order-theoretic meet-semilattice gives rise to a binary

´Ë µ

operation , and is a meet-semilattice in the algebraic sense as well. This is because (1) the

Ü Ü Ü Ü Ü

reﬂexitivity Ü implies that is a lower bound of . That is the greatest lower bound

ÝÜ Ü Ü

follows from the fact that there does not exist Ý with which is a lower bound of . That

Ü Ü Ü Ý Ü Ý

is, the idempotency law Ü holds. (2) Since the greatest lower bound of is

Ý Ü Ü Ý Ý Ü

also the greatest lower bound of , implying the commutativity law . (3) For

Ü ´Ý Þ µ Ü Ý Þ Û Ü

associativity, since Û is the greatest lower boud of and ,wehave and

Ý Þ Ý Þ Ý Ý Þ Þ Û Ý Û Þ

Û . Since , by transitivity we have . It follows that

Ü Ý Û Þ Û ´Ü Ý µ Þ Û ´Ü Ý µ Þ

Û and , hence . Similarly we can show that , hence

´Ý Þ µ´Ü Ý µ Þ

the associativity law Ü .

Ë µ

Conversely, if ´ is a meet-semilattice in the algebraic sense, then we can deﬁne

Ý Ü Ü Ý

Ü iff

Ý Ë

for all elements Ü and in . Now can check that (a) this relation does deﬁne a partial ordering in

´Ë µ ´Ë µ

Ë , (b) gives wrise to a meet-semilattice, and (c) the smeet-semilattice coincide with

Ë µ

the meet-semilattice ´ .

Ü Ü Ü Ü Ü Ý Ý Ü

(a) The idempotency Ü implies that refelexitivity .If then

Ü Ý Ý Ý Ü Ü Ý Ý Ü Ü Ý

Ü . Thus commutativity implies anti-symmetry .For

Ý Ý Þ Ü Ü Ý Ý Ý Þ

transitivity, suppose that Ü . Then we have and . So transitivity

Þ Ü Ü Þ

Ü , that is , follows

Ü Ü Ý Ü ´Ý Þ µ´Ü Ý µ Þ µÜ Þ

from assoiativity. This proves that is a partial order.

Ë µ Ü Ý ¾ Ë (b) To show that ´ is a meet-semilattice, we show that for any there exists a

greatest lower bound in Ë . By the idempotency and associativity, we have

Ü Ü ´Ü Ý µ ´Ü Üµ Ý Ü Ý

Ü Ý Ý Ü Ý Ü Ý Ü Ý

implying Ü . Similarly, .So is a lower bound of . Further, for any

Ü Ý Þ Ü Þ Ý Þ Þ Ü Þ Þ Ý

lower boud Þ of we have ,or , implying

Þ Þ Ü ´Þ Üµ Ý Þ ´Ü Ý µ

Ü Ý Ü Ý Ü Ý

by associativity. Hence Þ ,so is the (unique) greatest lower bound of . This

Ë µ

proves that ´ is a meet-semilattice.

Ý Ü Ý

Ë µ ´Ë µ the meet-semilattice ´ coincides with the original semi-lattice . Hence, the two deﬁnitions can be used in an entirely interchangeable way, depending on which of them appears to be more convenient for a particular purpose. A dual conclusion holds for join- semilattices.

2.2 Morphisms of semilattices

Ë µ ´Ì µ

Given two join-semilattices ´ and , a homomorphisms of (join-) semilattices is a func-

Ë Ì

tion with the property that

´Ü Ý µ ´Üµ ´Ý µ

That is, is a homomorphism of the two semigroups. If the join-semilattices are furthermore equipped with a least element ¼, then should also be a morphism of monoids, i.e. one additionally

requires that

´¼µ ¼ In the order-theoretical formulation, these conditions just state that a homomorphism of join- semilattices is a function that preserves binary joins and in the latter case also least elements. The conditions for homomorphisms of meet-semilattices are the obvious duals of these deﬁnitions. Note that any homomorphism of (both join- and meet-) semilattices is necessarily monotone

with respect to the associated ordering relation, that is

Ü Ý µ ´Üµ ´Ý µ

Or equivalently

Ü Ü Ý µ ´Üµ ´Üµ ´Ý µ

This follows from

´Üµ ´Ý µ ´Ü Ý µ ´Üµ

2.3 Free semilattices

3 Lattices

Lattices can be characterized both as posets and as algebraic structures. Both approaches are equivalent.

3.1 Lattices as posets

Ä µ Ü Ý Ä

DEFINITION 3.1 (LATTICE). A poset ´ is a lattice if for all elements and of , the set

Ü Ý has both a least upper bound (join,orsupremum) and a greatest lower bound (meet,or

inﬁmum).

Ý Ü Ý Ü Ý

The join and meet of Ü and are denoted by and , respectively. Because joins and meets are assumed to exist in a lattice, and are binary operations. The deﬁnition is equivalent

to requiring Ä to be both a meet- and a join-semilattice.

References

[1] Donnellan, Thomas, 1968. Lattice Theory. Pergamon. [2] Gratzer, G., 1971. Lattice Theory: First concepts and distributive lattices. W. H. Freeman. [3] Davey, B.A., and H. A. Priestley, 2002. Introduction to Lattices and Order. Cambridge Uni- versity Press. [4] Garrett Birkhoff, 1967. Lattice Theory, 3rd ed. Vol. 25 of American Mathematical Society Colloquium Publications. American Mathematical Society. [5] Johnstone, P.T., 1982. Stone spaces. Cambridge Studies in Advanced Mathematics 3. Cam- bridge University Press.