If H is a graph with vertex set {[v.sub.1], [v.sub.2], ..., [v.sub.k]}, then a homomorphism of G to H is an ordered partition ([V.sub.1], [V.sub.2], ..., [V.sub.k]) of V(G), with some cells allowed to be empty, such that an edge of G can have one end in [V.sub.i] and the other in [V.sub.j] only if [v.sub.i][v.sub.j] [member of] E(H).

A conditional colouring with condition C and template H, also known as an (H, C)-partition, is an ordered partition ([V.sub.1], [V.sub.2], ..., [V.sub.k]) of V(G) into sets which each induce a subgraph belonging to the family C, such that edge of G can have one end in Vi and the other in [V.sub.j] only if [v.sub.i][v.sub.j] [member of] E (H).

Suppose that L is a lattice and ([A.sub.1], [A.sub.2], ..., [A.sub.n]) is an ordered partition of the atoms of L.

Definition 7 Let L be a lattice and let ([A.sub.1], [A.sub.2], ..., [A.sub.n]) be an ordered partition of the atoms of L.

An

ordered partition (or partition for short) of V is a list [pi] = ([W.sub.1], ..., [W.sub.m]) of nonempty pairwise disjoint subsets of V whose union is V.

A face F which has dimension d corresponds to an ordered partition of [r] into d parts according to Ziegler (1995).

By reordering the coordinates, we may assume that F corresponds to the ordered partition ([1..[i.sub.1]], [[i.sub.1] + + [i.sub.2]], ..., [[i.sub.1] + ...

One can say more for words where only two letters occur since an ordered partition with two parts is determined by the subset I = {[i.sub.1], [i.sub.2], ..., [i.sub.s]} of [n] that appears in its second row, and is denoted accordingly by [bar.I].

5.5.7] each [pi] [member of] [[DELTA].sup.r.sub.n] may be written uniquely in the form of a special ordered partition, called the tabloid form of [pi], which is defined as

The empty face corresponds to the ordered partition ([n]).

[[alpha].sub.n] in the symmetric group [G.sub.n] and a composition [??] = ([c.sub.1], ..., [c.sub.k]) of n, define the ordered partition