Proposition 3.2 Every IA-continuous map is an IA?-continuous map.
Proof Let (X, ?) and (X?, ??) be an ITSs and V be ??-IRO set in X?. Since f : X ? X? is an IA-continuous map, f -1(V ) is a ?-IO set in X. Again since each IO set is an I?-open set( see [1]), f -1(V ) is a ?-I? open set in X. Thus f is an IA?-continuous map.
Remark 3.2 The converse of the above result is not necessarily true.
Proposition 3.3 Every IA?-continuous map is an IA?-continuous.
Proof Let (X, ?) and (X?, ??) be an ITSs and V be ??-IRO set in X?. Since f : X ? X? is an IA?-continuous map, f -1(V ) is a ?-I?-open set in X. Again since each I?-open set is an I?-open set( see [1]), f -1(V ) is a ?-I?-open set in X. Thus f is an IA?-continuous map.
Remark 3.3 The converse of the above result is not necessarily true.
Example 3.7 Let X and X? be ITSs where X=X?= {a,b,c} and ? = { ?~, X~, , }, ?? = = { ??~ , X?~, , } on X and X? respectively. We consider a map f: X? X? defined by f(a)=b, f(b)=a, f(c)=c. We see that ??~ , X?~, and are ??-IRO set in X? but f -1(??~) = ?~, f -1(X?~ ) = X~ , f -1() = and f -1() = are ?-I?-open sets but not a ?-I? open sets in X. Thus f is an IA?-continuous map but not an IA? -continuous map.
Proposition 3.4 Every IA?-continuous map is an IA?-continuous.
Proof Let (X, ?) and (X?, ??) be an ITSs and V be ??-IRO set in X?. Since f : X ? X? is an IA?-continuous map, f -1(V ) is a ?-I?-open set in X. Again since each I?-open set is an I?-open set( see [1]), f -1(V ) is a ?-I?-open set in X. Thus f is an IA?-continuous map.
Remark 3.4 The converse of the above result is not
Example 3.8 Let X and X? be ITSs where X=X?= {a,b,c} and ? = { ?~, X~, , }, ?? = { ??~ , X?~, , } on X and X? respectively. We consider a map f: X? X? defined by f(a)=b, f(b)=a, f(c)=c. We see that ??~ , X?~, and are ??-IRO set in X? but f -1(??~) = ?~, f -1(X?~ ) = X~ , f -1() = and f -1() = are ?-I?-open sets but not a ?-I? open sets in X. Thus f is an IA?-continuous map but not an IA? -continuous map.
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