Fig. 2. All rules with two elements of Ki with 05 support, 50% confidence As we want to analyze all facets of the folksonomy, we want to allow to use any of the three sets U, T, and R as the set of objects- on which the support is computed- at some point in time, depending on the task on hand Therefore. we will not fix the roles of the three sets in advance. Instead, we consider a triadic context as symmetric structure, where all three sets are of equal importance. For easier handling, we will therefore denote the folksonomy F:=(U,T,R, Y alternatively by F: =(X1, X2, X3, Y) in the following We will define the set of objects -i. e, the set on which the support will be counted- by a permutation on the set 1, 2, 3, i. e, by an element o of the full symmetric group S3. The choice of a permutation indicates, together with one of the aggregation modes 9,'W, n' with n E N, and"V, on which formal context K: =(G, M, I the association rules are computed ·K:=(Xa(1)×Xa(3),Xa(),D)with(x(1),xr(3),xa(2)∈ I if and only if( ·Ko:=(Xa(1),Xa(2)×X(3),D)with(x(1),(xa(2),x(3)∈ I if and only if(x1,x2,x3)∈Y Ko,n: =(Xa(), Xo(2), I)with(aa(1), Ta(2)E I if and only if there exist n different z(3)∈Xa(3)with(x1,x2,r3)∈Y K, V : =(Xa(), Xa(2), I)with(aa(1), Ta(2)E I if and only if for all To(3)E Xo(3) holds(=1, r2, r3)E Y. The mode "V is thus equivalent to 'n' if These projections are complemented by the following way to ' cut slices'out of the folksonomy. A slice is obtained by selecting one dimension (out of user/tag/resource), and then fixing in this dimension one particular instance ·Ietx:=x(3)∈X(3Kox:=(Xa(1),Xa(2), D) with(xa(),x(2)∈Iif and only if (a1, r2, I3)EY In the next section, we will discuss for a selected subset of these projections the kind of rules one obtains from mining the formal context that is resulting from the projection 5 Mining Association Rules on the Projected Folksonomy After having performed one of the projections described in the previous sec- tion, one can now apply the standard association rule mining techniques asFig. 2. All rules with two elements of K1 with .05 % support, 50 % confidence As we want to analyze all facets of the folksonomy, we want to allow to use any of the three sets U, T , and R as the set of objects – on which the support is computed – at some point in time, depending on the task on hand. Therefore, we will not fix the roles of the three sets in advance. Instead, we consider a triadic context as symmetric structure, where all three sets are of equal importance. For easier handling, we will therefore denote the folksonomy F := (U, T, R, Y ) alternatively by F := (X1, X2, X3, Y ) in the following. We will define the set of objects – i. e., the set on which the support will be counted – by a permutation on the set {1, 2, 3}, i. e., by an element σ of the full symmetric group S3. The choice of a permutation indicates, together with one of the aggregation modes ‘ G ’, ‘ M ’, ‘∃n’ with n ∈ N, and ‘∀’, on which formal context K := (G, M, I) the association rules are computed. • Kσ, G := (Xσ(1) × Xσ(3), Xσ(2), I) with ((xσ(1), xσ(3)), xσ(2)) ∈ I if and only if (x1, x2, x3) ∈ Y . • Kσ, M := (Xσ(1), Xσ(2) × Xσ(3), I) with (xσ(1),(xσ(2), xσ(3))) ∈ I if and only if (x1, x2, x3) ∈ Y . • Kσ,∃n := (Xσ(1), Xσ(2), I) with (xσ(1), xσ(2)) ∈ I if and only if there exist n different xσ(3) ∈ Xσ(3) with (x1, x2, x3) ∈ Y . • Kσ,∀ := (Xσ(1), Xσ(2), I) with (xσ(1), xσ(2)) ∈ I if and only if for all xσ(3) ∈ Xσ(3) holds (x1, x2, x3) ∈ Y . The mode ‘∀’ is thus equivalent to ‘∃n’ if |Xσ(3)| = n. These projections are complemented by the following way to ‘cut slices’ out of the folksonomy. A slice is obtained by selecting one dimension (out of user/tag/resource), and then fixing in this dimension one particular instance. • Let x := xσ(3) ∈ Xσ(3). Kσ,x := (Xσ(1), Xσ(2), I) with (xσ(1), xσ(2)) ∈ I if and only if (x1, x2, x3) ∈ Y . In the next section, we will discuss for a selected subset of these projections the kind of rules one obtains from mining the formal context that is resulting from the projection. 5 Mining Association Rules on the Projected Folksonomy After having performed one of the projections described in the previous sec￾tion, one can now apply the standard association rule mining techniques as 6