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In Mathematics , a lattice is a Partially Ordered Set (or '' Poset '') whose nonempty ''finite'' Subset s all have a Supremum (called '''join''') and an Infimum (called '''meet'''). Lattices can also be characterized as Algebraic Structures satisfying certain axiomatic Identities . Since the two definitions are equivalent, lattice theory draws on both Order Theory and Universal Algebra . Semilattice s include lattices, which in turn include Heyting and Boolean Algebra s. These "lattice-like" structures all admit order-theoretic as well as algebraic descriptions. LATTICES AS POSETS Consider a Poset (''L'', ≤). ''L'' is a lattice if : For all elements ''x'' and ''y'' of ''L'', the set {''x'', ''y''} has both a least upper bound (join, or ''supremum'') and a greatest lower bound ('''meet''', or ''infimum''). The join and meet of ''x'' and ''y'' are denoted by and , respectively. Because joins and meets are assumed to exist in a lattice, and are Binary Operation s. Hence this definition is equivalent to requiring ''L'' to be both a Meet- and a Join-semilattice . A bounded lattice has a Greatest and Least Element , denoted 1 and 0 by convention (also called '''top''' and '''bottom'''). Any lattice can be converted into a bounded lattice by adding a greatest and least element. Using an easy Induction argument, one can deduce the existence of suprema (''joins'') and infima (''meets'') of all non-empty finite subsets of any lattice. With additional assumptions, further conclusions may be possible; see the article on Completeness In Order Theory for more discussion of this subject. That article also discusses how one may rephrase the above definition in terms of the existence of suitable Galois Connection s between related posets — an approach of special interest for the Category Theoretic approach to lattices. LATTICES AS ALGEBRAIC STRUCTURES Let ''L'' be a Set with two binary Operations , and . A ''lattice'' is an Algebraic Structure of type , such that the following axiomatic Identities hold for all members ''a'', ''b'', and ''c'' of ''L'': The following important identity follows from the above: These axioms assert that (''L'',) and (''L'',) are each Semilattice s. The absorption laws, the only equations in which both meet and join appear, distinguish a lattice from a pair of semilattices and assure that the two semilattices interact appropriately. In particular, each semilattice is the Dual of the other. A ''bounded lattice'' requires that meet and join each have a neutral element, called 1 and 0 by convention. See the entry Semilattice . Lattices have some connections to the Groupoid family. Because meet and join both commute and associate, a lattice can be viewed as consisting of two commutative Semigroups having the same carrier. If a lattice is bounded, these semigroups are also commutative Monoid s. The Absorption Law is the only defining identity that is peculiar to lattice theory. The are the least and greatest elements, respectively. Therefore a lattice contains all finite (including empty) meets and joins only if it is bounded. For this reason, some authors define a lattice so as to require that 0 and 1 be members of of ''L''. While definining a lattice in this manner entails no loss of generality, because any lattice can be embedded in a bounded lattice, this definition will not be adopted here. The algebraic interpretation of lattices plays an essential role in Universal Algebra . CONNECTION BETWEEN THE TWO DEFINITIONS The algebraic definition of a lattice implies the order theoretic one, and vice versa. Obviously, an order-theoretic lattice gives rise to two binary operations and . It is easy to see that these operations make (''L'', , ) into a lattice in the algebraic sense. The converse is true also: Consider an algebraically defined lattice (''M'', , ). Now define a partial order ≤ on ''M'' by setting : ''x'' ≤ ''y'' Iff ''x'' = ''x''''y'' or, equivalently, : ''x'' ≤ ''y'' iff ''y'' = ''x''''y'' for elements ''x'' and ''y'' in ''M''. The laws of absorption ensure that both definitions are indeed equivalent. 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 and . Conversely, the order induced by the algebraically defined lattice (''L'', , ) that was derived from the order theoretic formulation above coincides with the original ordering of ''L''. Since the two definitions of a lattice are equivalent, one may freely invoke aspects of either definition in any way that suits the purpose at hand. EXAMPLES
Further examples are given for each of the additional properties discussed below. MORPHISMS OF LATTICES The appropriate notion of a Morphism between two lattices flows easily from the above algebraic definition. Given two lattices (''L'', , ) and (''M'', , ), a ''homomorphism of lattices'' is a function ''f'' : ''L'' → ''M'' such that : ''f''(''x''''y'') = ''f''(''x'') ''f''(''y''), and : ''f''(''x''''y'') = ''f''(''x'') ''f''(''y''). Thus ''f'' is a Homomorphism of the two underlying Semilattice s. If the lattices are bounded, then ''f'' should preserve the bounds so that: : ''f''(0) = 0, and : ''f''(1) = 1. In the order-theoretic formulation, these conditions just state that a homomorphism of lattices is a function Preserving binary meets and joins. For bounded lattices, preservation of least and greatest elements is just preservation of join and meet of the empty set. Any homomorphism of lattices is necessarily Monotone with respect to the associated ordering relation; see Preservation Of Limits . The converse is of course not true: monotonicity by no means implies the required preservation properties. Given the standard definition of Isomorphism s as invertible morphisms, a lattice isomorphism is just a Bijective lattice homomorphism. Lattices and their homomorphisms form a Category . PROPERTIES OF LATTICES We now introduce a number of important properties that lead to interesting special classes of lattices. One, boundedness, has already been discussed. Completeness A highly relevant class of lattices are the Complete Lattice s. A lattice is complete if ''all'' of its subsets have both a join and a meet, which should be contrasted to the above definition of a lattice where one only requires the existence of all (non-empty) ''finite'' joins and meets. Details can be found within the respective article. Distributivity Since any lattice comes with two binary operations, it is natural to consider whether one Distributes over the other. A lattice (''L'', , ) is distributive, if the following condition is satisfied for every three elements ''x'', ''y'' and ''z'' of ''L'': : This condition is equivalent to the Dual statement: : Other characterizations exist, and can be found in the article on Distributive Lattice s. For complete lattices one can formulate various stronger properties, giving rise to the classes of Frames and Completely Distributive Lattice s. For an overview of these different notions, see Distributivity In Order Theory . Modularity Distributivity is too strong a condition for certain applications. A strictly weaker property is ''modularity'': a lattice (''L'', , ) is Modular if, for all elements ''x'', ''y'', and ''z'' of ''L'', we have : Another equivalent statement of this condition is as follows: if ''x'' ≤ ''z'' then for all ''y'' one has : For example, the lattice of submodules of a module, and the lattice of Normal Subgroup s of a group, all have this special property. Moreover, every distributive lattice is modular. Continuity and algebraicity In Domain Theory , it is natural to seek approximating the elements in a partial order by "much simpler" elements. This leads to the class of Continuous Poset s, consisting of posets where any element can be obtained as the supremum of a Directed Set of elements that are Way-below the element. If one can additionally restrict these to the Compact Element s of a poset for obtaining these directed sets, then the poset is even Algebraic . Both concepts can be applied to lattices as follows:
Both of these classes have interesting properties. For example, continuous lattices can be characterized as algebraic structures (with infinitary operations) satisfying certain identities. While such a characterization is not known for algebraic lattices, they can be described "syntactically" via Scott Information System s. Complements and pseudo-complements Let ''L'' be a bounded lattice with greatest element 1 and least element 0. Two elements ''x'' and ''y'' of ''L'' are complements of each other iff: : and In this case, we write ¬''x'' = ''y'' and equivalently, ¬''y'' = ''x''. A bounded lattice for which every element has a complement is called a Complemented Lattice . The corresponding unary Operation over ''L'', called Complementation , introduces an analogue of logical Negation into lattice theory. The complement is not necessarily unique, nor does it have a special status among all possible unary operations over ''L''. A complemented lattice that is also distributive is a Boolean Algebra . For a Boolean Algebra , the complement of ''x'' is provably unique. Heyting Algebra s are an example of distributive lattices having at least some members lacking complements. Every element ''x'' of a Heyting algebra has, on the other hand, a ''pseudo-complement'', also denoted ¬''x''. The pseudo-complement is the greatest element ''y'' such that ''x''''y'' = 0. If the pseudo-complement of every element of a Heyting algebra is in fact a complement, then the Heyting algebra is in fact a Boolean algebra. FREE LATTICES Using the standard definition of from the Category of sets and functions to the category of lattices and lattice homomorphisms which is Left Adjoint to the Forgetful Functor from lattices to their underlying sets. We treat the case of bounded lattices, i.e. algebraic structures with the two binary operations and and the two constants (nullary operations) 0 and 1. The set of all correct (well-formed) expressions that can be formulated using these operations on elements from a given set of generators ''S'' will be called W(''S''). This set of words contains many expressions that turn out to be equal in any lattice. For example, if ''a'' is some element of ''S'', then ''a''1 = 1 and ''a''1 =''a''. The ''word problem'' for lattices is the question, which of these elements have to be identified. The answer to this problem is as follows. Define a relation <~ on W(''S'') by setting ''w'' <~ ''v'' Iff one of the following holds:
This defines a Preorder <~ on W(''S''). The partially ordered set induced by this preorder (i.e. the set obtained by identifying all words ''w'' and ''v'' with ''w''<~''v'' and ''v''<~''w'') is the free lattice on ''S''. The required embedding ''i'' is the obvious mapping from a generator ''a'' to (the set of words equivalent to) the word ''a''. One of the consequences of this statement is that the free lattice of a three element set of generators is already infinite. In fact, one can even show that every free lattice on three generators contains a sublattice which is free for a set of four generators. By Induction this eventually yields a sublattice free on Countably many generators. The case of lattices that are not bounded is treated similarly, using only the two binary operations in the above construction. IMPORTANT LATTICE-THEORETIC NOTIONS In the following, let ''L'' be a lattice. We define some order-theoretic notions that are of particular importance in lattice theory. An element ''x'' of ''L'' is called join-irreducible iff
When the first condition is generalized to arbitrary joins Vai, ''x'' is called completely join-irreducible. The dual notion is called meet-irreducibility. Sometimes one also uses the terms v-irreducible and ^-irreducible, respectively. An element ''x'' of ''L'' is called join-prime iff
Again, this can be generalized to obtain the notion completely join-prime and dualized to yield '''meet-prime'''. Any join-prime element is also join-irreducible, and any meet-prime element is also meet-irreducible. If the lattice is distributive the converse is also true. Other important notions in lattice theory are Ideal and its dual notion Filter . Both terms describe special subsets of a lattice (or of any partially ordered set in general). Details can be found in the respective articles. SEE ALSO REFERENCES A monograph available free online:
Elementary texts recommended for those with limited mathematical maturity:
The standard contemporary introductory text:
The classic advanced monograph:
Free lattices are discussed in the following title, not primarily devoted to lattice theory:
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