An Introduction to the Theory of Lattice

Transcription

1 An Introduction to the Theory of Lattice Jinfang Wang Λy Graduate School of Science and Technology, Chiba University 1-33 Yayoi-cho, Inage-ku, Chiba , Japan May 11, 2006 Λ Fax: addresses: y Partially supported by Grand-in-Aid of the Japanese Ministry of Education, Science, Sports and Culture. 1

2 1 Introduction A lattice 1 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 DEFINITION 2.1 (MEET-SEMILATTICE). A poset (S;») is a meet-semilattice if for all elements x and y of S, the greatest lower bound (meet) of the set fx; yg exists. Dually, we define a poset (S;») as a join-semilattice, if for all elements x and y of S, the least upper bound (join) of the set fx; yg exists. The join and meet of x and y are denoted by x _ y and x ^ y, 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. Semilattices as algebraic structures. Consider an algebraic structure given by (S; ^), ^ being a binary operation. DEFINITION 2.2 (MEET-SEMILATTICE). A poset (S; ^) is a meet-semilattice if the following identities hold for all elements x, y, and z in S: x ^ (y ^ z) =(x ^ y) ^ z (associativity) x ^ y = y ^ x x ^ x = x (commutativity) (idempotency) 1 The term lattice derives from the shape of the Hasse diagrams that result from depicting the orders. 2

3 In this case, ^ is called meet. A join-semilattice is an algebraic structure (S; _), where the meet _ satisfies the same axioms as above, the difference in notation being only for convenience, usually for dual interpretations in specific applications. Definition 2.2 says that (S; ^) is a semilattice if (S; ^) is an idempotent, commutative semigroup. Sometimes it is desirable to consider meet-, join-semilattices with greatest (least) elements, which are defined as idempotent, commutative monoids. Explicitly, (S; ^; 1) are meet-semilattices with greatestelements if (S; ^) are meet-semilattices and in addition we have x ^ 1=x for all x 2 S. Similarly, (S; _; 0) are join-semilattices with least elements if (S; _) are joinsemilattices and x _ 0=x for all x 2 S. Equivalence of both definitions. An order-theoretic meet-semilattice gives rise to a binary operation ^, and (S; ^) is a meet-semilattice in the algebraic sense as well. This is because (1) the reflexitivity x» x implies that x is a lower bound of fx; xg. That x is the greatest lower bound follows from the fact that there does not exist y with y>xwhich is a lower bound of fx; xg. That is, the idempotency law x ^ x = x holds. (2) Since the greatest lower bound x ^ y of fx; yg is also the greatest lower bound of fy; xg, implying the commutativity law x ^ y = y ^ x. (3) For associativity, since w = x ^ (y ^ z) is the greatest lower boud of x and y ^ z, wehavew» x and w» y ^ z. Since y ^ z» y; y ^ z» z, by transitivity we have w» y; w» z. It follows that w» x ^ y and w» z, hence w» (x ^ y) ^ z. Similarly we can show that w (x ^ y) ^ z, hence the associativity law x ^ (y ^ z) =(x ^ y) ^ z. Conversely, if (S; ^) is a meet-semilattice in the algebraic sense, then we can define x» y iff x = x ^ y for all elements x and y in S. Now can check that (a) this relation does define a partial ordering in S, (b) (S;») gives wrise to a meet-semilattice, and (c) the smeet-semilattice (S;») coincide with the meet-semilattice (S; ^). (a) The idempotency x = x ^ x implies that refelexitivity x» x. If x» y; y» x then x = x ^ y; y = y ^ x. Thus commutativity x ^ y = y ^ x implies anti-symmetry x = y. For transitivity, suppose that x» y; y» z. Then we have x = x ^ y and y = y ^ z. So transitivity x» z, that is x = x ^ z, follows x = x ^ y = x ^ (y ^ z) =(x ^ y) ^ z) =x ^ z from assoiativity. This proves that» is a partial order. (b) To show that (S;») is a meet-semilattice, we show that for any x; y 2 S there exists a greatest lower bound in S. By the idempotency and associativity, we have x = x ^ (x ^ y) =(x ^ x) ^ y = x ^ y 3

4 implying x» x ^ y. Similarly, y» x ^ y. Sox ^ y is a lower bound of fx; yg. Further, for any lower boud z of fx; yg we have z» x; z» y,orz = z ^ x; z = z ^ y, implying z = z ^ x =(z ^ x) ^ y = z ^ (x ^ y) by associativity. Hence z» x ^ y, sox ^ y is the (unique) greatest lower bound of fx; yg. This proves that (S;») is a meet-semilattice. (c) We have seen in (b) that x ^ y is the unique greatest lower bound of fx; yg, showing that the meet-semilattice (S;») coincides with the original semi-lattice (S; ^). 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 joinsemilattices. 2.2 Morphisms of semilattices Given two join-semilattices (S; _) and (T;_), a homomorphisms of (join-) semilattices is a function f : S! T with the property that f (x _ y) =f (x) _ f (y) : That is, f is a homomorphism of the two semigroups. If the join-semilattices are furthermore equipped with a least element 0, then f should also be a morphism of monoids, i.e. one additionally requires that f (0) = 0 : In the order-theoretical formulation, these conditions just state that a homomorphism of joinsemilattices 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 x» y =) f (x)» f (y) Or equivalently This follows from x = x _ y =) f (x) =f (x) _ f (y) f (x) _ f (y) =f (x _ y) =f (x) 2.3 Free semilattices 3 Lattices Lattices can be characterized both as posets and as algebraic structures. Both approaches are equivalent. 4

5 3.1 Lattices as posets DEFINITION 3.1 (LATTICE). A poset (L;») is a lattice if for all elements x and y of L, the set fx; yg has both a least upper bound (join, orsupremum) and a greatest lower bound (meet, or infimum). The join and meet of x and y are denoted by x _ y and x ^ y, respectively. Because joins and meets are assumed to exist in a lattice, _ and ^ are binary operations. The definition is equivalent to requiring L to be both a meet- and a join-semilattice. References [1] Donnellan, Thomas, Lattice Theory. Pergamon. [2] Gratzer, G., Lattice Theory: First concepts and distributive lattices. W. H. Freeman. [3] Davey, B.A., and H. A. Priestley, Introduction to Lattices and Order. Cambridge University Press. [4] Garrett Birkhoff, Lattice Theory, 3rd ed. Vol. 25 of American Mathematical Society Colloquium Publications. American Mathematical Society. [5] Johnstone, P.T., Stone spaces. Cambridge Studies in Advanced Mathematics 3. Cambridge University Press. 5