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In Mathematics , a lattice is a Partially Ordered Set (or ''poset'') in which every pair of elements has a unique Supremum (the elements' least upper bound; called their ''' Join ''') and an Infimum (greatest lower bound; called their ''' 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 Join- and a Meet-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'' Completeness (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 axioms above 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 greatest and least 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 ''L''. While defining 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'' If And Only If ''x'' = ''x''''y'' or, equivalently, : ''x'' ≤ ''y'' if and only if ''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'' or ''lattice homomorphism'' 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. When lattices with more structure are considered, the morphisms should 'respect' the extra structure, too. Thus, a morphism ''f'' between two bounded lattices ''L'' and ''M'' should also have the following property: : ''f''(0''L'') = 0''M'' , and : ''f''(1''L'') = 1''M'' . 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. Similarly, a ''lattice endomorphism'' is a lattice homomorphism from a lattice to itself, and a ''lattice automorphism'' is a bijective lattice endomorphism. 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 to approximate 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 if and only if: : 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 distributive lattice, the complement of ''x'' when it exists 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. SUBLATTICES A ''sublattice'' of a lattice ''L'' is a nonempty subset of ''L'' which is a lattice with the same meet and join operations as ''L''. That is, if ''L'' is a lattice and ''M'' is a subset of ''L'' such that for every pair of elements ''a'', ''b'' in ''M'' both ''a''''b'' and ''a''''b'' are in ''M'', then ''M'' is a sublattice of ''L''.Burris, Stanley N., and H.P. Sankappanavar, H. P., 1981. '' A Course in Universal Algebra. '' Springer-Verlag. ISBN 3-540-90578-2. A sublattice ''M'' of a lattice ''L'' is a ''convex sublattice'' of ''L'', if ''x ≤ z ≤ y'' and ''x'', ''y'' in ''M'' implies that ''z'' belongs to ''M'', for all elements ''x, y, z'' in ''L''. FREE LATTICES See Also: Free lattice Any set ''X'' may be used to generate the free semilattice ''FX''. The free semilattice is defined to consist of all of the finite subsets of ''X'', with the semilattice operation given by ordinary Set Union . The free semilattice has the Universal Property . 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 if and only if
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 if and only if
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. An element ''x'' of ''L'' is an atom, if ''L'' has a ''0'', ''0 < x'', and there exists no element ''y'' of ''L'' such that ''0 < y < x''. We say that ''L'' is '''atomic''', if every nonzero element of ''L'' lies above some atom of ''L''. We say that ''L'' is '''atomistic''', if every element of ''L'' is a Supremum of atoms, that is, for all ''a, b'' in ''L'' such that , there exists an atom ''x'' of ''L'' such that and . 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 Monographs 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:
The standard textbook on free lattices:
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