![]() So it is not a bounded lattice in general. This lattice has the empty set as least element, but it will only contain a greatest element if A itself is finite. The lattice operations are intersection (meet) and union (join) of sets, respectively. ![]() For any set A, the collection of all finite subsets of A (including the empty set) can be ordered via subset inclusion to obtain a 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. Conversely, the order induced by the algebraically defined lattice ( L, \vee, \wedge) that was derived from the order theoretic formulation above coincides with the original ordering of L. 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 \vee and \wedge. The above laws for absorption ensure that both definitions are indeed equivalent. Now one can define a partial order ≤ on M by settingįor all elements x and y in M. Maybe more surprisingly, one can also obtain the converse of this result: consider any algebraically defined lattice ( M, \vee, \wedge). It now can be seen very easily that this operation really makes ( L, \vee, \wedge) a lattice in the algebraic sense. Obviously, an order theoretic lattice gives rise to two binary operations \vee and \wedge. For details compare the article on semilattices. In order to describe bounded lattices, one has to include neutral elements 0 and 1 for the meet and join operations in the above definition. Furthermore, it turns out that the idempotency laws can be deduced from absorption and thus need not be stated separately. Note that the laws for idempotency, commutativity, and associativity just state that ( L,\vee) and ( L,\wedge) constitute two semilattices, while the absorption laws guarantee that both of these structures interact appropriately. a \wedge (b \wedge c) = (a \wedge b) \wedge c L is a lattice if the following identities hold for all elements a, b, and c in L: This article also discusses how one may rephrase the above definition in terms of the existence of suitable Galois connections between related posets - an approach that is of special interest for category theoretic investigations of the concept.Ĭonsider an algebraic structure in the sense of universal algebra, given by ( L,\vee, \wedge), where \vee and \wedge are two binary operations. See the article on completeness in order theory for more discussion on this subject. Further conclusions may be possible in the presence of other properties. Using an easy induction argument, one can also conclude the existence of all suprema and infima of non-empty finite subsets of any lattice. If both of these special elements do exist, then the poset is a bounded lattice. It will be stated explicitly whenever a lattice is required to have a least or greatest element. Also note that the above definition is equivalent to requiring L to be both a meet- and a join-semilattice. ![]() ![]() Clearly, this defines binary operations \wedge and \vee on lattices. In this situation, the join and meet of x and y are denoted by x\vee y and x\wedge y, respectively. L is a lattice ifįor all elements x and y of L, the set has both a least upper bound ( join) and a greatest lower bound ( meet). Both approaches and their relationship are explained below.Ĭonsider a poset ( L, ≤). As mentioned above, lattices can be characterized both as posets and as algebraic structures.
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