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| In [[mathematics]], in the area of [[order theory]], a '''free lattice''' is the [[free object]] corresponding to a [[Lattice (order)|lattice]]. As free objects, they have the [[universal property]].
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| ==Formal definition==
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| 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]] {''x''}. 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 <math>\tilde{f}:FX\to L</math> such that <math>f=\tilde{f}\circ\eta</math>. The map <math>\tilde{f}</math> may be explicitly written down; it is given by
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| :<math>S\in FX \mapsto\bigvee\left\{f(s)\vert s\in S\right\}</math>
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| Here, <math>\bigvee</math> denotes the semilattice operation in ''L''. This construction may be promoted from semilattices to '''lattices'''{{clarify|reason=Give the construction of a free lattice explicitly. If it consists in using finite subsets of X with union and intersection, it will produce a distributive lattice. In that case, no homomorphism into a non-distributive lattice can exist.|date=September 2013}}; by construction the map <math>\tilde{f}</math> will have the same properties as the lattice.
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| The symbol ''F'' is then a [[functor]] from the [[category of sets]] to the category of lattices and lattice homomorphisms. The functor ''F'' is [[adjoint functors|left adjoint]] to the [[forgetful functor]] from lattices to their underlying sets. The free lattice is a [[free object]].
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| ==Word problem==
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| {| style="float:right; border: 1px solid #808080"
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| |+ Example computation of ''x''∧''z'' ~ ''x''∧''z''∧(''x''∨''y'')
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| {| style=" border: 1px solid #808080"
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| | align="right"|''x''∧''z''∧(''x''∨''y'') || ≤<sub>~</sub> || ''x''∧''z''
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| | by 5.
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| | since
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| | align="right"|''x''∧''z'' || ≤<sub>~</sub> || ''x''∧''z''
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| | by 1.
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| | since
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| | align="right"|''x''∧''z'' || = || ''x''∧''z''
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| |}
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| {| style=" border: 1px solid #808080"
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| | align="right"| ''x''∧''z'' || ≤<sub>~</sub> || ''x''∧''z''∧(''x''∨''y'')
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| |-
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| | by 7.
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| | since
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| | align="right"|''x''∧''z'' || ≤<sub>~</sub> || ''x''∧''z''
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| | and
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| | align="right"|''x''∧''z'' || ≤<sub>~</sub> || ''x''∨''y''
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| |-
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| | by 1.
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| | since
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| | align="right"|''x''∧''z'' || = || ''x''∧''z''
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| | by 6.
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| | since
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| | align="right"|''x''∧''z'' || ≤<sub>~</sub> || ''x''
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| | by 5.
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| | since
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| | align="right"|''x'' || ≤<sub>~</sub> || ''x''
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| | by 1.
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| | since
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| | align="right"|''x'' || = || ''x''
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| |}
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| |}
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| The [[word problem (mathematics)|word problem]] for free lattices has some interesting aspects. Consider the case of bounded lattices, i.e. algebraic structures with the two binary operations ∨ and ∧ and the two constants ([[nullary operation]]s) 0 and 1. The set of all well-formed [[term (logic)|expressions]] that can be formulated using these operations on elements from a given set of generators ''X'' will be called '''W'''(''X''). This set of words contains many expressions that turn out to denote equal values in every lattice. For example, if ''a'' is some element of ''X'', then ''a''∨1 = 1 and ''a''∧1 =''a''. The '''word problem''' for free bounded lattices is the problem of determining which of these elements of '''W'''(''X'') denote the same element in the free bounded lattice ''FX'', and hence in every bounded lattice.
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| The word problem may be resolved as follows. A relation ≤<sub>~</sub> on '''W'''(''X'') may be defined [[mathematical induction|inductively]] by setting ''w'' ≤<sub>~</sub> ''v'' [[if and only if]] one of the following holds:
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| # ''w'' = ''v'' (this can be restricted to the case where ''w'' and ''v'' are elements of ''S''),
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| # ''w'' = 0,
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| # ''v'' = 1,
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| # ''w'' = ''w''<sub>1</sub> ∨ ''w''<sub>2</sub> and both ''w''<sub>1</sub>≤<sub>~</sub>''v'' and ''w''<sub>2</sub>≤<sub>~</sub>''v'' hold,
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| # ''w'' = ''w''<sub>1</sub> ∧ ''w''<sub>2</sub> and either ''w''<sub>1</sub>≤<sub>~</sub>''v'' or ''w''<sub>2</sub>≤<sub>~</sub>''v'' holds,
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| # ''v'' = ''v''<sub>1</sub> ∨ ''v''<sub>2</sub> and either ''w''≤<sub>~</sub>''v''<sub>1</sub> or ''w''≤<sub>~</sub>''v''<sub>2</sub> holds,
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| # ''v'' = ''v''<sub>1</sub> ∧ ''v''<sub>2</sub> and both ''w''≤<sub>~</sub>''v''<sub>1</sub> and ''w''≤<sub>~</sub>''v''<sub>2</sub> hold.
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| This defines a [[preorder]] ≤<sub>~</sub> on '''W'''(''X''), so an [[equivalence relation]] can be defined by ''w''~''v'' when ''w''≤<sub>~</sub>''v'' and ''v''≤<sub>~</sub>''w''. One may then show that the [[Partially ordered set|partially ordered]] [[quotient space]] '''W'''(''X'')/~ is the free bounded lattice ''FX''.<ref>P. Whitman, [http://dx.doi.org/10.2307/1969001 "Free Lattices"], ''Ann. Math.'' '''42''' (1941) pp. 325–329</ref><ref>P. Whitman, [http://dx.doi.org/10.2307/1968883 "Free Lattices II"], ''Ann. Math.'' '''43''' (1941) pp. 104–115</ref> The [[equivalence class]]es of '''W'''(''X'')/~ are the sets of all words ''w'' and ''v'' with ''w''≤<sub>~</sub>''v'' and ''v''≤<sub>~</sub>''w''. Two well-formed words ''v'' and ''w'' in '''W'''(''X'') denote the same value in every bounded lattice if and only if ''w''≤<sub>~</sub>''v'' and ''v''≤<sub>~</sub>''w''; the latter conditions can be effectively decided using the above inductive definition. The table shows an example computation to show that the words ''x''∧''z'' and ''x''∧''z''∧(''x''∨''y'') denote the same value in every bounded lattice. The case of lattices that are not bounded is treated similarly, omitting rules 2. and 3. in the above construction.
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| The solution of the word problem on free lattices has several interesting corollaries. One is that the free lattice of a three-element set of generators is 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 [[mathematical induction|induction]], this eventually yields a sublattice free on [[countable|countably]] many generators.<ref>L.A. Skornjakov, ''Elements of Lattice Theory'' (1977) Adam Hilger Ltd. ''(see pp.77-78)''</ref> This property is reminiscent of [[SQ-universality]] in [[group (mathematics)|groups]].
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| The proof that the free lattice in three generators is infinite proceeds by inductively defining
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| :''p''<sub>''n''+1</sub> = ''x'' ∨ (''y'' ∧ (''z'' ∨ (''x'' ∧ (''y'' ∨ (''z'' ∧ ''p''<sub>''n''</sub>)))))
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| where ''x'', ''y'', and ''z'' are the three generators, and ''p''<sub>0</sub>=''x''. One then shows, using the inductive relations of the word problem, that ''p''<sub>''n''+1</sub> is strictly greater<ref>that is, ''p''<sub>''n''</sub> ≤<sub>~</sub> ''p''<sub>''n''+1</sub>, but not ''p''<sub>''n''+1</sub> ≤<sub>~</sub> ''p''<sub>''n''</sub></ref>
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| than ''p''<sub>''n''</sub>, and therefore all infinitely many words ''p''<sub>''n''</sub> evaluate to different values in the free lattice ''FX''.
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| ==The complete free lattice==
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| Another corollary is that the [[complete free lattice]] "does not exist", in the sense that it is instead a [[proper class]]. The proof of this follows from the word problem as well. To define a [[complete lattice]] in terms of relations, it does not suffice to use the [[finitary relation]]s of [[meet and join]]; one must also have [[infinitary relation]]s defining the meet and join of infinite subsets. For example, the infinitary relation corresponding to "join" may be defined as
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| :<math>\operatorname{sup}_N:(f:N\to FX)</math>
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| Here, ''f'' is a map from the elements of a [[Cardinal number|cardinal]] ''N'' to ''FX''; the operator <math>\operatorname{sup}_N</math> denotes the supremum, in that it takes the image of ''f'' to its join. This is, of course, identical to "join" when ''N'' is a finite number; the point of this definition is to define join as a relation, even when ''N'' is an infinite cardinal.
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| The axioms of the pre-ordering of the word problem may be adjoined by the two infinitary operators corresponding to meet and join. After doing so, one then extends the definition of <math>p_n</math> to an [[ordinal number|ordinally]] indexed <math>p_\alpha</math> given by
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| :<math>p_\alpha = \operatorname{sup}\{p_\beta \vert \beta<\alpha \}</math>
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| when <math>\alpha</math> is a [[limit ordinal]]. Then, as before, one may show that <math>p_{\alpha+1}</math> is strictly greater than <math>p_\alpha</math>. Thus, there are at least as many elements in the complete free lattice as there are ordinals, and thus, the complete free lattice cannot exist as a set, and must therefore be a proper class.
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| ==References==
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| {{reflist}}
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| * Peter T. Johnstone, ''Stone Spaces'', Cambridge Studies in Advanced Mathematics 3, Cambridge University Press, Cambridge, 1982. (ISBN 0-521-23893-5) ''(See chapter 1)''
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| [[Category:Lattice theory]]
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| [[Category:Free algebraic structures]]
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| [[Category:Combinatorics on words]]
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Friends call her Claude Gulledge. Playing crochet is a thing that I'm totally addicted to. Her family lives in Idaho. I am a cashier and I'll be promoted soon.
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