diff --git a/graph/check_test.yaml b/graph/check_test.yaml
index c5a17cb..89577b9 100644
--- a/graph/check_test.yaml
+++ b/graph/check_test.yaml
@@ -15,6 +15,10 @@ types:
   user:
     relations:
       manager: user
+      friend: group#member
+    permissions:
+      in_management_chain: manager | manager->in_management_chain
+      unfriendly_manager: in_management_chain - friend
 
   ### display_name: Identity ###
   identity:
diff --git a/graph/objects_test.go b/graph/objects_test.go
index 0332968..c0249e8 100644
--- a/graph/objects_test.go
+++ b/graph/objects_test.go
@@ -23,13 +23,18 @@ func TestSearchObjects(t *testing.T) {
 	require.NoError(err)
 	require.NotNil(m)
 
-	mnfst := manifest(m.Invert())
+	im := m.Invert()
+	mnfst := manifest(im)
 
 	b, err := yaml.Marshal(mnfst)
 	require.NoError(err)
 
 	t.Logf("inverted model:\n%s\n", b)
 
+	require.NoError(
+		im.Validate(model.SkipNameValidation, model.AllowPermissionInArrowBase),
+	)
+
 	for _, test := range searchObjectsTests {
 		t.Run(test.search, func(tt *testing.T) {
 			assert := assert.New(tt)
diff --git a/model/inverse.go b/model/inverse.go
index 748f30c..6050511 100644
--- a/model/inverse.go
+++ b/model/inverse.go
@@ -3,6 +3,7 @@ package model
 import (
 	"fmt"
 
+	set "github.com/deckarep/golang-set/v2"
 	"github.com/samber/lo"
 )
 
@@ -94,8 +95,28 @@ func (i *inverter) applySubstitutions(o *Object) {
 }
 
 func (i *inverter) invertPermission(on ObjectName, pn RelationName, o *Object, p *Permission) {
+	var typeSet relSet
+
+	switch {
+	case p.IsUnion():
+		typeSet = set.NewSet(p.Types()...)
+	case p.IsIntersection():
+		typeSet = lo.Reduce(p.Terms(), func(acc relSet, pt *PermissionTerm, i int) relSet {
+			s := set.NewSet(pt.Types()...)
+			if i == 0 {
+				return s
+			}
+			if s.IsEmpty() {
+				return acc
+			}
+			return acc.Intersect(s)
+		}, nil)
+	case p.IsExclusion():
+		typeSet = set.NewSet(p.Exclusion.Include.Types()...)
+	}
+
 	for _, pt := range p.Terms() {
-		newTermInverter(i, on, pn, o, p, pt).invert()
+		newTermInverter(i, on, pn, o, p, pt, typeSet).invert()
 	}
 }
 
@@ -122,10 +143,11 @@ type termInverter struct {
 	obj      *Object
 	perm     *Permission
 	term     *PermissionTerm
+	typeSet  relSet
 	kind     permissionKind
 }
 
-func newTermInverter(i *inverter, on ObjectName, pn RelationName, o *Object, p *Permission, pt *PermissionTerm) *termInverter {
+func newTermInverter(i *inverter, on ObjectName, pn RelationName, o *Object, p *Permission, pt *PermissionTerm, typeSet relSet) *termInverter {
 	return &termInverter{
 		inv:      i,
 		objName:  on,
@@ -133,6 +155,7 @@ func newTermInverter(i *inverter, on ObjectName, pn RelationName, o *Object, p *
 		obj:      o,
 		perm:     p,
 		term:     pt,
+		typeSet:  typeSet,
 		kind:     kindOf(p),
 	}
 }
@@ -155,7 +178,8 @@ func (ti *termInverter) invertArrow() {
 	itip := ti.inv.irelSub(ti.objName, ti.term.Base)
 	baseRel := ti.obj.Relations[ti.term.Base]
 
-	for _, rr := range ti.perm.Types() {
+	typeRefs := set.NewSet(ti.perm.Types()...)
+	typeRefs.Intersect(ti.typeSet).Each(func(rr RelationRef) bool {
 		iName := InverseRelation(ti.objName, ti.permName, rr.Relation)
 		iPerm := permissionOrNew(ti.inv.im.Objects[rr.Object], iName, ti.kind)
 		for _, baseRR := range baseRel.Types() {
@@ -165,24 +189,28 @@ func (ti *termInverter) invertArrow() {
 			rel := relationOrNew(ti.inv.im.Objects[baseRR.Object], itip)
 			rel.AddRef(&RelationRef{Object: baseRR.Object, Relation: itip})
 		}
-	}
+		return false // resume iteration
+	})
 }
 
 func (ti *termInverter) invertRelation() {
-	for _, rr := range ti.obj.Relations[ti.term.RelOrPerm].Types() {
+	typeRefs := set.NewSet(ti.obj.Relations[ti.term.RelOrPerm].Types()...)
+	typeRefs.Intersect(ti.typeSet).Each(func(rr RelationRef) bool {
 		iName := InverseRelation(ti.objName, ti.permName, rr.Relation)
 		ip := permissionOrNew(ti.inv.im.Objects[rr.Object], iName, ti.kind)
 		ip.AddTerm(&PermissionTerm{RelOrPerm: ti.inv.irelSub(ti.objName, ti.term.RelOrPerm, rr.Relation)})
-	}
+		return false // resume iteration
+	})
 }
 
 func (ti *termInverter) invertPermission() {
-	for _, rr := range types(ti.obj, ti.term.RelOrPerm) {
+	typeRefs := set.NewSet(types(ti.obj, ti.term.RelOrPerm)...)
+	typeRefs.Intersect(ti.typeSet).Each(func(rr RelationRef) bool {
 		iName := InverseRelation(ti.objName, ti.permName, rr.Relation)
 		ip := permissionOrNew(ti.inv.im.Objects[rr.Object], iName, ti.kind)
 		ip.AddTerm(&PermissionTerm{RelOrPerm: ti.inv.irelSub(ti.objName, ti.term.RelOrPerm, rr.Relation)})
-
-	}
+		return false // resume iteration
+	})
 }
 
 type permissionKind int
diff --git a/model/types.go b/model/types.go
index 6493758..2e10dbd 100644
--- a/model/types.go
+++ b/model/types.go
@@ -215,12 +215,7 @@ func (p *Permission) AddTerm(pt *PermissionTerm) {
 }
 
 func (p *Permission) Types() RelationRefs {
-	return append(
-		lo.Map(p.SubjectTypes, func(on ObjectName, _ int) RelationRef {
-			return RelationRef{Object: on}
-		}),
-		p.Intermediates...,
-	)
+	return append(objectNamesToRelationRefs(p.SubjectTypes), p.Intermediates...)
 }
 
 type PermissionTerm struct {
@@ -243,6 +238,13 @@ func (pr *PermissionTerm) IsArrow() bool {
 	return pr.Base != ""
 }
 
+func (pr *PermissionTerm) Types() RelationRefs {
+	return append(
+		objectNamesToRelationRefs(pr.SubjectTypes),
+		pr.Intermediates...,
+	)
+}
+
 type PermissionTerms []*PermissionTerm
 
 func (pts PermissionTerms) Contains(pt *PermissionTerm) bool {
@@ -254,6 +256,12 @@ func (pts PermissionTerms) Contains(pt *PermissionTerm) bool {
 	return false
 }
 
+func objectNamesToRelationRefs(names []ObjectName) RelationRefs {
+	return lo.Map(names, func(on ObjectName, _ int) RelationRef {
+		return RelationRef{Object: on}
+	})
+}
+
 type ExclusionPermission struct {
 	Include *PermissionTerm `json:"include,omitempty"`
 	Exclude *PermissionTerm `json:"exclude,omitempty"`