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In mathematics, in the area of order theory, a free lattice is the free object corresponding to a lattice. As free objects, they have the universal property. ==Formal definition== 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. The universal morphism is (''FX'',η), where η is the unit map η:''X''→''FX'' which takes ''x''∈''X'' to the singleton set . The universal property is then as follows: given any map ''f'':''X''→''L'' from ''X'' to some arbitrary semilattice ''L'', there exists a unique semilattice homomorphism such that . The map may be explicitly written down; it is given by : Here, denotes the semilattice operation in ''L''. This construction may be promoted from semilattices to lattices; by construction the map will have the same properties as the lattice. The symbol ''F'' is then a functor from the category of sets to the category of lattices and lattice homomorphisms. The functor ''F'' is left adjoint to the forgetful functor from lattices to their underlying sets. The free lattice is a free object. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「free lattice」の詳細全文を読む スポンサード リンク
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