Free lattice


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


Because the concept of a lattice can be axiomatised in terms of two operations   and   satisfying certain identities, the category of all lattices constitute a variety (universal algebra), and thus there exist (by general principles of universal algebra) free objects within this category: lattices where only those relations hold which follow from the general axioms.

These free lattices may be characterised using the relevant universal property. Concretely, free lattice is a functor   from sets to lattices, assigning to each set   the free lattice   equipped with a set map   assigning to each   the corresponding element  . The universal property of these is that there for any map   from   to some arbitrary lattice   exists a unique lattice homomorphism   satisfying  , or as a commutative diagram:   The functor   is left adjoint to the forgetful functor from lattices to their underlying sets.

It is frequently possible to prove things about the free lattice directly using the universal property, but such arguments tend to be rather abstract, so a concrete construction provides a valuable alternative presentation.



In the case of semilattices, an explicit construction of the free semilattice   is straightforward to give; this helps illustrate several features of the definition by way of universal property. Concretely, the free semilattice   may be realised as the set of all finite nonempty subsets of  , with ordinary set union as the join operation  . The map   maps elements of   to singleton sets, i.e.,   for all  . For any semilattice   and any set map  , the corresponding universal morphism   is given by   where   denotes the semilattice operation in  .

This form of   is forced by the universal property: any   can be written as a finite union of elements on the form   for some  , the equality in the universal property says  , and finally the homomorphism status of   implies   for all  . Any extension of   to infinite subsets of   (if there even is one) need however not be uniquely determined by these conditions, so there cannot in   be any elements corresponding to infinite subsets of  .

Lower semilattices


It is similarly possible to define a free functor   for lower semilattices, but the combination   fails to produce the free lattice   in several ways, because   treats   as just a set:

  • the join operation   is not extended to the new elements of  ,
  • the existing partial order on   is not respected;   views   and   as unrelated, not understanding it should make  .

The actual structure of the free lattice   is considerably more intricate than that of the free semilattice.

Word problem

Example computation of xz ~ xz∧(xy)
xz∧(xy) ~ xz
by 5. since xz ~ xz
by 1. since xz = xz
xz ~ xz∧(xy)
by 7. since xz ~ xz and xz ~ xy
by 1. since xz = xz by 6. since xz ~ x
by 5. since x ~ x
by 1. since x = x

The 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 operations) 0 and 1. The set of all well-formed 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.

The word problem may be resolved as follows. A relation ≤~ on W(X) may be defined inductively by setting w~ v if and only if one of the following holds:

  1.   w = v (this can be restricted to the case where w and v are elements of X),
  2.   w = 0,
  3.   v = 1,
  4.   w = w1w2 and both w1~ v and w2~ v hold,
  5.   w = w1w2 and either w1~ v or w2~ v holds,
  6.   v = v1v2 and either w~ v1 or w~ v2 holds,
  7.   v = v1v2 and both w~ v1 and w~ v2 hold.

This defines a preorder~ on W(X), so an equivalence relation can be defined by w ~ v when w~ v and v~ w. One may then show that the partially ordered quotient space W(X)/~ is the free bounded lattice FX.[1][2] The equivalence classes of W(X)/~ are the sets of all words w and v with w~ v and v~ w. Two well-formed words v and w in W(X) denote the same value in every bounded lattice if and only if w~ v and v~ w; the latter conditions can be effectively decided using the above inductive definition. The table shows an example computation to show that the words xz and xz∧(xy) 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.

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 induction, this eventually yields a sublattice free on countably many generators.[3] This property is reminiscent of SQ-universality in groups.

The proof that the free lattice in three generators is infinite proceeds by inductively defining

pn+1 = x ∨ (y ∧ (z ∨ (x ∧ (y ∨ (zpn)))))

where x, y, and z are the three generators, and p0 = x. One then shows, using the inductive relations of the word problem, that pn+1 is strictly greater[4] than pn, and therefore all infinitely many words pn evaluate to different values in the free lattice FX.

The complete free lattice


Another corollary is that the complete free lattice (on three or more generators) "does not exist", in the sense that it is 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 relations of meet and join; one must also have infinitary relations defining the meet and join of infinite subsets. For example, the infinitary relation corresponding to "join" may be defined as


Here, f is a map from the elements of a cardinal N to FX; the operator   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.

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   to an ordinally indexed   given by


when   is a limit ordinal. Then, as before, one may show that   is strictly greater than  . 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.


  1. ^ Philip M. Whitman, "Free Lattices", Ann. Math. 42 (1941) pp. 325–329
  2. ^ Philip M. Whitman, "Free Lattices II", Ann. Math. 43 (1941) pp. 104–115
  3. ^ L.A. Skornjakov, Elements of Lattice Theory (1977) Adam Hilger Ltd. (see pp.77-78)
  4. ^ that is, pn~ pn+1, but not pn+1~ pn
  • Peter T. Johnstone, Stone Spaces, Cambridge Studies in Advanced Mathematics 3, Cambridge University Press, Cambridge, 1982. (ISBN 0-521-23893-5) (See chapter 1)