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An Analysis of Structural Validity of Ternary Relationshipsin Entity Relationship Modeling

James Dullea and Il-Yeol SongCollege of Information Science and Technology

Drexel UniversityPhiladelphia, Pennsylvania 19104

Email: james.dullea@boeing.com and songiy@post.drexel.edu

Abstract____________________________________________________________________________________________________

This research explores the criteria that contribute to the validity of modeling structures within the entity-relationship (ER) diagram. Our analysis yields a comprehensive set of decision rules to determine the structuralvalidity of any ternary relationship in an ER diagram. These rules can be readily applied to real world modelsregardless of the complexity of the problem. They can easily be incorporated into the database modeling anddesigning process or extended into case tool implementations. In previous approaches only maximumcardinality constraints were used in their analysis. Here we use both maximum and minimum cardinalityconstraints yielding a more complete analysis of the structural validity of ternary relationship types. Our studyalso includes an analysis of ternary relationships and unrelated binary relationships. Previous studies onlyperformed standalone validity evaluations of ternary relationships. In our study we evaluate the ternaryrelationship as part of the overall diagram and our rules address ternary relationships as they coexist with otherrelationships in a path structure within the model. Our approach examines the embedded binary relationshipsthat exist between any two entities in a ternary relationship to determine the structural validity of the pathcontaining the ternary relationship. We investigate constraining binary relationships and those properties thatallow or disallow them to further define the ternary relationship. We further explore the effects of minimumcardinality constraints on constraining binary relationships and how they affect structural validity. Thesignificant contribution of this paper is to furnish a complete set of rules based on impacts of constrainingrelationships, path connectivity, and cardinality constraints to determine the structural validity of an entity-relationship diagram containing binary and ternary relationships.

1. Introduction

Entity-relationship (ER) modeling [CHEN76] is the foundation of various analysis and design

methodologies for the development of information systems and relational databases. A key measure

of success in the design of these models is the level to which they accurately reflect the real world

environment they intend to represent. A model can be a very complex abstract structure, and

designers are highly prone to making small mistakes that incorporate inconsistencies into the

structure. There is various supporting empirical evidence [BOEH76], [DALY77], [FAGA74] that

concludes errors, mistakes, or inconsistencies made in the early stages of the software development

life cycle are very expensive to correct. Boehm [BOEH81] states that the cost difference to correct

an error in the early phases as opposed to the post-implementation phase is on the order of a ratio of

one to one hundred. Left undetected these inconsistencies become very costly to correct, so early

discovery of an error is highly desirable. Direct treatment of structural validity as a concept is rare in

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the data modeling literature. Indirect treatment about structural validity is generally used to make

buttress points about other issues. Structural validity is almost always treated as a property of a

study; we intend to treat it as the object of our study. In this paper we perform a complete analysis of

the structural validity of ternary relationships as used in ER diagrams. Figure 1 shows a path

containing binary relationships and a constrained ternary relationship with their accompanying

semantics. This diagram looks plausible, but it is structurally invalid, and our work presented in this

paper will explain why. See Section 4.2.3 for the discussion on the validity of this particular diagram.

Figure 1: Invalid ERD: PROJECTS are IMPLEMENTED through a TEAM with BUDGETED funds. PROJECTSare constrained to being FUNDED through only one BUDGET. PROJECTS CONTRACT with CONSULTING

FIRMS for services. CONSULTANTS WORKING FOR CONSULTING FIRMS SUPPORT the TEAMS.

BUDGET IMPLEMENTS TEAM

PROJECT

M

11

CONTRACTWITH

M

M

FUNDEDM

1

CONSULTINGFIRM

CONSULTANTSSUPPORTS

WORKSFOR

1

11

M

The purpose of this research is to investigate structural validity in the entity-relationship

model and develop a comprehensive set of rules that will allow designers to identify modeling

structure inconsistencies that yield invalid structures. This research proposes to perform a complete

exploration of structural validity in ER diagramming. In this paper we investigate and analyze ternary

structures. We consider both minimum and maximum cardinality constraints to develop a set of

heuristics that can determine the validity of any acyclic or cyclic path containing ternary relationships

in an ER diagram and identify those elements that contribute to invalidity.

This paper is organized into five sections. In Section Two the concepts of semantic and

structural validity, both necessary for a valid design, are discussed and defined. Section Three

summarizes our work on the structural validity of binary relationships. Six rules are presented and

briefly explained. These rules will play an important role in the analysis of ternary relationships.

Section Four explores the effects of ternary relationship structures on the validity of the diagram. We

extensively explore the embedded binary relationships derived from the ternary relationship to

perform our path analysis. Ternary relationships are also examined for standalone validity and the

redundancy [DULL97] of embedded binary relationships in conjunction with imposed constraining

binary relationships. Both single and multiple constraining relationships are explored. In addition, we

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consider the effects of minimum cardinality on constraining binary relationships. Section Five states

the conclusions of our research, summarizes the rules, and identifies future research areas.

2. Validity

An entity-relationship model is composed of entities, the relationships between entities, and

constraints on those relationships. Entities may be chained together in a series of alternating entities

and relationships. This series of alternating entities and relationships is called a path. Paths are the

building blocks of our study in structural validity analysis, and visually define the semantic and

structural association that each entity has simultaneously with all other entities within the path. We

define the terms structural and semantic validity as follows:

Definitions: An Entity Relationship Diagram is structurally valid only when its paths can

support all of the structural constraints imposed on the model simultaneously over all possible

intended states. An Entity Relationship Diagram is semantically valid only when each and every

path exactly represents the modeler’s concept of the problem domain. An Entity Relationship

Diagram is valid when it is both semantically and structurally valid.

In data modeling validity can be classified into two types: semantic and structural validity.

Semantic validity is a micro view of the model. Each individual relational concept between entities

along with the accompanying constraints is represented in a diagrammatical form. The coupling of

these concepts develop a structure represented as a diagram. This diagram must communicate

exactly the intended concept of the environment as viewed by the modeler.

Structural validity is a macro view of the model. There are two types of structural

constraining factors imposed on a model: maximum cardinality and minimum cardinality. The driving

force behind cardinality constraint placement is the semantics of the model. The values and placement

of these constraints must be robust enough to convey the business rules exactly intended by the

modeler while being consistent with respect to the whole model in order to reflect the real world

environment. The model is not just a set of individually constrained relationships pieced together

between sets of entities but a holistic view of the representation domain. Each set of cardinality

constraints on a single relationship must be consistent with all the remaining constraints in the model

and over all possible states. This constraint compliant state considers optional participation for

minimum cardinality to be always at least one instance not participating in the relationship and the

‘MANY’ constraint for maximum cardinality to be always at least two instances mapping to a single

instance in the relationship.

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In order for a model to be valid all the paths in the model must also be valid. Our analysis will

investigate those types of structural paths in an entity-relationship diagram that are critical to validity

of the entire diagram. Our study will focus primarily on the structural validity of the ternary

relationships represented in an ER diagram

3. Binary Relationships

A binary relationship is an association between the instances of one entity with the instances

of another entity. These two entities may participate in other relationships simultaneously including

being member entities of higher degree relationships. Binary relationships may occur in conjunction

with a ternary relationship in three different ways. First, an embedded binary relationship exists

between any two entities in a ternary relationship through the association of each entity’s instances

(Figure 2a). Second, a binary relationship may exist between two of the three entities and be

unrelated to the concept being communicated by the ternary relationship(Figure 2b). Third, a simple

ternary relationship may not fully define the concept intended to be modeled. In this case a

constraining binary relationship on the ternary is required to further define the concept(Figure 2c).

Figure 2a: Embedded binaryrelationships exist

PART SUPPLY PROJECT

SUPPLIER

M 11

M

NM

MN N

Figure 2b: READS is unrelated toOWNERSHIP

STORE OWNERSHIP BOOK

PERSON

M 11

READSM

1

Figure 2c: INSURED BY constraintsHEALTH_SERVICE

DOCTOR HEALTHSERVICE HMO

PATIENT

M 1N

INSUREDBYM

1

The structural validity of a ternary relationship is dependent upon the correct participation of

the ternary relationship with other relationships in the diagram and the proper imposition of any

constraining binary relationships on the ternary relationship. Because of the underlying dependency

on binary relationship, we will summarize an earlier study in the binary structural validity presented in

another paper [DULL98].

From that study we established six basic rules from our analysis of binary relationships. These

rules will be extensively in our analysis of ternary relationships because only two of entities in a

ternary participate in a path. There are two types of paths in a diagram: Acyclic and cyclic. Acyclic

paths are open ended and do not recur back on previous entities. The simplest acyclic path is a simple

binary relationship involving only two entities. A more complex acyclic path is made up of the

coupling of additional entities to previously existing entities through the relationship mechanism. A

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cyclic path is a closed path having the capability of starting with and ending with the same entity.

Because it is a closed path the instance composition of any given entity is sometimes dependent on

the relationship with other entities not immediately adjacent to it in the path. Table 1 states each rule

with an example diagram of a valid structure.

RULES VALID EXAMPLE1. An acyclic path containing all binary relationships is always

structurally valid.V R(vw) W

X

1 1

M

R(yz)

R(xy)

R(wx)Z

Y

1

M

M

M

1

2. A cyclic path containing all ‘One-to-One’ Mandatory-Mandatorybinary relationships is always structurally valid.

X

R(xz)

Z

1

1R(xy)

R(yz)Y

1

1

1

1

3. A cyclic path that contains all binary relationships, and one or more‘Optional-Optional’ relationships is always structurally valid.

X

R(xz)

Z

M

1R(xy)

R(yz)Y

1

1

1

M

4. A cyclic path that contains all binary relationships, and one or more‘Many-to-One’ relationships with ‘Optional’ participation on the ‘One’side is always structurally valid.

X

R(xz)

Z

M

1R(xy)

R(yz)Y

1

1

1

M

5. A cyclic path that contains all binary relationships, and one or more‘Many-to-Many’ relationships is always structurally valid.

X

R(xz)

Z

M

1R(xy)

R(yz)Y

1

N

1

M

6. Cyclic paths containing at least one binary relationship that is[{‘One-to-Many’, Totally Mandatory}, {‘One-to-Many’, Mandatory-Optional}, or {‘One-to-One’, Mandatory-Optional}] and at least oneother binary relationship that is [{‘Many-to-One’, Totally Mandatory},{‘Many-to-One’, Optional-Mandatory }, or {‘One-to-One’, Optional-Mandatory}] is structurally valid.

X

R(xz)

Z

M

1R(xy)

R(yz)Y

M

1

1

M

Table 1. Structural Validity Rules for Binary RelationshipsThe structural validity of an ER diagram containing all binary relationships is dependent on its

paths supporting the structural constraints imposed by the model simultaneously over all possible

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intended states. The simplest acyclic path is always valid because the instance composition (number

of instances)of each entity required by the semantics is always supported by this simple structural

path. The instance composition of any given entity in an acyclic path is only dependent on the

relationship with the entities immediately adjacent to it and not any other entity-relationship in the

path. Since an acyclic path is really a group of simple relationships chained together independent of

other entities then the acyclic path is always structurally valid [Rule 1].

The remaining five rules on binary relationships address cyclic paths and the dependency they

create. The situation involving Rule 2 rarely occurs in the real world. If it occurs it is structurally

valid because in a series of 1:1 mandatory-mandatory relationships the instance composition for any

entity is always equal to every other entity in the cyclic path [Rule 2].

There are certain types of relationships that for structural purposes act as a self-adjusting

relationship with respect to the instance composition of its entities. A self-adjusting relationship

exists when the combination of the minimum and maximum cardinality constraints allow the instance

composition of the entities to vary freely without restriction. A ‘One to Many’ Mandatory-

Mandatory relationship is very restrictive in the sense that the instance composition on the ‘One’ side

must always be less than the instance composition on the ‘Many’ side. If we change the minimum

cardinality constraints to Optional-Optional, then most restrictions on the instance composition of the

participating entities disappear. The only restriction that remains is that the number of instances on

the ‘One’ side must be greater than zero and on the ‘Many’ side greater than one. A cyclic path

containing at least one ‘Optional-Optional’ relationship is always structurally valid because the

structure is self-adjusting can support any set of valid semantics regardless of the remaining

cardinality constraints [Rule 3]. In a ‘Many to One’ relationship only the ‘One’ side has to be

‘Optional’ to remove the limitations on the instance composition of the entities [Rule 4]. If the

‘Optional’ cardinality constraint is imposed only on the ‘Many’ side the inequality placed on the

instance composition of the entities remain in effect. When the ‘Optional’ constraint is imposed on

the ‘One’ side regardless of the maximum cardinality constraint of the ‘Many’ side the inequality is

removed and allows the instances of the two entities to adjust to meet the semantic requirements of

the model. The ‘Many to Many’ relationship is also self-adjusting relationship. The instance

composition of each entity in a ‘Many to Many’ relationship is not encumbered by an inequality

restriction. Each entity may contain any number of instances independent of the other entity and

therefore confirms to the required semantics of the model [Rule 5].

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The combination of two opposing relationships in a diagram can have the same self-adjusting

effect as the relationships in Rules 3, 4, and 5. In Table 2 any cardinality constraint shown in Column

1 imposes a greater than inequality between the entities (reading left to right) while those in Column

2 imposes a less than inequality between the entities. The presents in a cyclic path of any relationship

with the cardinality constraints in Column 1 in conjunction with a relationship with the cardinality

constraints in Column 2 removes the inequality restrictions from the diagram [Rule 6].

Greater Than Less Than

Many-to-One’, Mandatory-Mandatory ‘One-to-Many’, Totally Mandatory

Many-to-One’, Optional-Mandatory ‘One-to-Many’, Mandatory-Optional

One-to-One’, Optional-Mandatory ‘One-to-One’, Mandatory-Optional

Table 2: Opposing Relationships

The rules presented in this summary will be used extensively in our analysis of ternary relationships.

4. The Effects of Ternary Relationships on Validity

In section 4 we will examine the criteria to evaluate the validity of an ER diagram containing

paths with binary and ternary relationships. Ternary relationships are associations involving three

entities. In a ternary relationship the association of an instance from each entity participating in the

relationship is represented as a triple [ELMA94] [TEOR94]. Membership in the triple implies that

instance pairs of two entities is associated with an instance from the remaining entity. Consider the

example of a ternary relationship with entities X, Y, and Z, each containing one instance X1, Y1, and

Z1, respectively. The triple represents the association of the instance pair (Y1, Z1) with X1, (X1, Z1)

with Y1, and (X1, Y1) with Z1. Continuing to populate the entities with instances, the number of

triples can increase by the combinations of allowable ‘instance pairs to instances’ in the relationship.

The allowable pairing is only restricted by the maximum and minimum cardinality constraints on the

relationship.

In our tables and diagrams we will be using ‘One (1) and Many (M)’ notation for maximum

cardinality and the symbol “#” to indicate mandatory minimum cardinality and the symbol “)” to

indicate optional minimum cardinality [SONG95].

4.1 Ternary Relationships as a Member of an Acyclic Paths

Like binary relationships, ternary relationships can be part of an acyclic or cyclic path. In a

ternary relationship participating in either an acyclic or cyclic path only two of the entities participate

in the path. When evaluating the validity of an acyclic path containing a ternary relationship, only the

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binary relationship between the two entities in the path is used to evaluate the path. The binary

relationship between two entities of a ternary relationship is derived from the ternary relationship and

can either be explicit or implicit. Since our previous Rule 1 state that an acyclic path containing all

binary relationships is always valid and since a derived binary association represents the binary

relationship between two entities in a ternary relationship, then an acyclic path containing a ternary

relationship is always structurally valid. We can now represent the idea as a modification of Rule 1 to

include ternary relationships.

Rule 1: If a path containing binary and ternary relationship is an acyclic path, then the path is

always structurally valid.

4.2 Ternary Relationships as a Member of a Cyclic Paths

[SONG93] and [JONE96] have performed comprehensive studies of the simultaneous

coexistence of ternary and binary relationships, and the interrelationships created by their joint

coexistence. They identified three types of binary-ternary coexistence relationships: Unconstrained

ternary relationships with derived binary relationships between entities, ternary relationships further

constrained by explicit binary relationships, and unrelated binary relationships between two entities of

a ternary relationship. In their analysis they established rules to determine the standalone validity of

these three relationship types. The purpose of the next three sections is to investigate and establish

rules to determine the overall validity of an ER diagram containing these three relationship types.

4.2.1 No Explicit Binary Relationships (Constraining or Unrelated) Between Entities

In this section we examine the criteria for evaluating the validity of cyclic path containing a ternary

relationship with no explicit binary relationships between entities. Between any two entities in a

ternary relationship, when there is no explicit constraining binary relationship, there is an implicit

binary relationship that describes the association between the instances of only these two entities.

[JONE96] established the Implicit Binary Cardinality (IBC) rule stated as follows:

“In any given ternary relationship, regardless of ternary cardinality, the implicit cardinalities

between any two entities must be considered ‘Many to Many’, provided that there is no explicit

restrictions on the number of instances that can occur.”

The following example demonstrates that the implicit binary relationship between any of the two

entities tends towards a ‘Many to Many’ relationship. The maximum cardinality constraints of a

ternary relationship have no effect on the binary relationship provided that there are no explicit

restrictions on the ternary relationship.

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A 1:1:M ternary relationship allows one of the entities instances to participate in the instance

pairings of the other entities more than once. The minimal set of triples to demonstrate a 1:1:M

relationship is:

X1:Y1:Z1 and X1:Y1:Z2

showing that the instance pair {X1,Y1} is associated with more that one instance in Entity Z. Adding

two additional triples (X2:Y1:Z3 and X1:Y2:Z3) as shown in Figure 3 will allow us to draw some

inference about maximum cardinality constraints of the embedded binary relationships. An embedded

binary relationship is a relationship between two entities in a ternary that is not explicitly modeled in

the diagram but the association that can be derived between two entities as they participate in the

ternary relationship. The observed binary relationships R(xy) between entities X and Y in the above

set of triples is

R(xyz)

X1:Y1:Z1X1:Y1:Z2X2:Y1:Z3X1:Y2:Z3

X R(xyz) Y

Z

1 1

M

N

M

M

M

N

N

Figure 3 : An unconstrained 1:1:M ternary relationship showing three embedded binary relationships

X Y Z

X1 Y1 Z1X2 Y2 Z2 Z3

Instance TablesR(xy)

X1:Y1

X2:Y1X1:Y2

R(xz)

X1:Z1X1:Z2X2:Z3X1:Z3

R(yz)

Y1:Z1Y1:Z2Y1:Z3Y2:Z3

also ‘Many-to-Many’, and likewise in entities X and Z from R(xz), and entities Y and Z from R(yz),

as shown in the instance tables in Figure 3 demonstrating that the maximum cardinality constraints for

the implicit binary relationships embedded in a 1:1:M ternary relationship tends towards M:N.

[JONE96] shows this phenomenon extends to all other combinations of maximum cardinality

constraints for ternary relationships.

The above example in Figure 3 considered the minimal number of instance triples to satisfy the

maximum cardinality constraints and show the cardinality of the embedded binary relationships tend

towards ‘Many to Many’. Adding two instances (Y9 and Z9) to entities Y and Z do not have any

effect on the maximum cardinality constraints of the embedded binary relationships as shown in the

instance tables in Figure 4. The conclusion is that the minimum cardinality constraints have no effect

on the IBC rule.

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R(xyz)

X1:Y1:Z1X1:Y1:Z2X2:Y1:Z3X1:Y2:Z3

X R(xyz) Y

Z

1 1

M

N

M

M

M

N

N

Figure 4 : An 1:1:M ternary relationship with minimun cardinaltiy constraints imposed showingther is no effect on the embedded binary relationships

X Y Z

X1 Y1 Z1X2 Y2 Z2 Z3 Y9 Z9

Instance TablesR(xy)

X1:Y1

X2:Y1X1:Y2

R(xz)

X1:Z1X1:Z2X2:Z3X1:Z3

R(yz)

Y1:Z1Y1:Z2Y1:Z3Y2:Z3

Y9 and Z9 do not participate in R(xyz)

We can now represent these ideas presented in sections 4.2.1 and 4.2.2 as a rule:

Rule 7: If a cyclic path contains a ternary relationship where there is no explicit restricting

binary relationships on the entities, then the path is always structurally valid, regardless of

maximum and minimum cardinality constraints on the ternary relationship.

4.2.2 The Effects on Structural Validity of Explicit Unrelated Binary Relationships involvedwith a Ternary Relationships

An explicit binary relationship between two entities participating in a ternary relationship can

be unrelated to the ternary relationship. When that binary relationship imposes a concept different

from the concept being presented by the ternary and does not constrain the instances participating in

the ternary relationship it is considered to be unrelated.

Figure 5b : A ternary relationship withan explicit unrelated binary relationship

Figure 5a : A Ternary Relationshipwith no explicit relationships

Store Owership Book

Person

Store Owership Book

Person Read

For example, if an association exists between the owner of a book, the title of the book, and store

where is the book is purchased, then a ternary relationship can be used to model this association.

Figure 5a shows how that relationship would be diagrammed. If we introduce an additional

relationship that is independent of the OWERSHIP relationship, such as the reader of the book, then

the binary relationship READ between Person and Book in Figure 5b is an explicit unrelated

relationship. Owning and reading a book are two different concepts. In an unrelated relationship the

maximum and minimum cardinality of the binary is independent of the ternary. From a structural

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validity view point unrelated binary relationships imposed on two entities in a ternary relationship

behave in the same manner as a ternary involved in a cyclic path and should be evaluate in the manner.

4.2.3 The Effects on Structural Validity of Explicit Constraining Binary Relationshipsimposed on Ternary Relationships

There are real world scenarios involving three entities that can not be modeled by using only a

simple ternary relationship because the implicit cardinality between any two entities is ‘Many to

Many’ but can be modeled using the combination of a ternary relationship and a constraining binary

relationship. A constraining binary relationship on the ternary relationship is used to further define

the association between two entities and is considered part of the ternary relationship. An explicit

constraining relationship restricts the instance groupings of the ternary relationship by directly

constraining the possible instance pairings allowed between the two entities. This concept is best

described as an example. Figure 6 shows a simple ternary relationship that associates PROJECT,

BUDGET, TEAM. We impose the following requirements on the example that will determine the

maximum cardinality of the relationship:

Projects are worked by teams with budgeted funds. Teams may work more than one project.

There are three Budgets (Research, Marketing, and Administrative) that may fund projects.

Teams can receive funding from multiple sources but a team working on a specific project may

only receive funding from one budget source for that project. Currently projects can receive

funding from multiple sources as long as each source is from a different team.

Figure 6 represents the above requirements in that each Project is associated with a Team and Budget

combination. The existence of the functional dependency {Budget, Project} à {Team}and {Team,

Project}à {Budget} from the requirements produces the M:1:1 ternary relationship. The instance

table in Figure 6 shows a possible set of instance triples that conform to the diagram and the

requirements. The instance table reflects the ‘Many to Many’ embedded binary relationships between

each pair of entities in the ternary relationship.

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BUDGET WORKS TEAM

PROJECT

1 1

MM

NM

MN M

Figure 6: A simple ternary relationship with the resulting embedded binary relationships and apossible set of instance triples

PROJECT12334556

BUDGETResarchResarchResarch

MarketingMarketing

AdminResarch

Marketing

TEAMT1T1T2T3T3T1T2T3

Instance Table

Because of a recent change in business policy on how projects are to funded an additional constraint

is imposed on our model.

Projects can only be funded from a single budget.

At first we may try to model this rule by modifying the ternary relationship. If we introduce the

functional dependency {Budget, Team} à {Project} we violate the condition that ‘Teams may work

more than one project’. Relaxing our constraints is certainly not the answer to our problem either.

Modifying the ternary association to create another simple ternary relationship does not accomplish

our objective. The reason is the new requirement’s association between Project and Budget is only a

binary association and does not directly involve the entity Team as shown by its functional

dependency {Project} à {Budget}. This is a constraining binary relationship that is considered part

of the ternary relationship because it is subset of the ternary relationship. {Project} à {Budget} is a

subset of {Team, Project} à {Budget}}. In a ternary relationship, a constraining binary relationship

can only be imposed where the binary’s FD is a subset of the ternary’s FD. [JONE96] established an

Explicit Binary Permission (EBP) Rule for evaluating potential combinations of explicit constraining

binary relationships allowed to be imposed on a ternary relationship. Their rule states: “For any

given ternary relationship, a binary relationship cannot be imposed where the binary cardinality is

less than the cardinality specified by the ternary, for any specific entity.”

Table 3 summarizes the allowable and disallowable binary impositions on the different cardinality

constraints of a ternary relationship.

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Allowed and Disallowed Constraining Binary Impositions1:1:1 Ternary Relationships

Any cardinality of a binary relationship can be imposed on a 1:1:1 ternary relationship1:1:M Ternary Relationships

X Y Z AllowedM N YESM 1 YES1 1 YES1 M YESM N YESM 1 NO1 1 NO1 M YES

M N YESM 1 NO1 1 NO1 M YES

1:M:N Ternary RelationshipsX Y Z AllowedM N YESM 1 NO1 1 NO1 M YESM N YESM 1 NO1 1 NO1 M YES

M N YESM 1 NO1 1 NO1 M NO

M:N:P Ternary RelationshipsOnly binary cardinality of M:N can be imposed on a M:N:P ternary relationshipThis imposition is redundant since the ternary relationship implicitly establishes the cardinality constraints

Table 3: Allowed and Disallowed Constraining Binary Impositions [JONE96]

The constraining binary relationship in our example is allowable according to the EBP rule. Figure 7

shows the ternary relationship from Figure 6 with the explicit binary relationship imposed, the

resulting maximum cardinality constraints of the embedded binary relationships, and a possible

instance table to reflect the instance triples of the diagram. Important to our analysis of structural

validity is the changes that occur to the embedded binary relationships due to the imposition. The

embedded relationship between Project and Budget follows the constraining M:1 relationship as

expected but additional the embedded relationship between Project and Team also changes to M:1

because of the imposition. The embedded relationships change because of the additional binary FD(s)

imposed on the ternary relationship and the additional binary FDs that may be derived from the

imposition on the ternary relationships.

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BUDGET WORKS TEAM

PROJECT

1 1

MM

NM

M1 1

PROJECT123456

BUDGETResearchResearchResearchMarketing

AdminMarketing

TEAMT1T1T2T3T1T3

Instance Table

FUNDEDM

1

Figure 7: A ternary relationship with an explicit constraining binary relationship and theresulting embedded binary relationships with a possible set of instance triples

[JONE96] analyzed the effects of an imposed constraining binary relationship on the embedded binary

relationships in a ternary relationship and their results are presented in Table 4.Ternary Relationship

X:Y:ZImposed Constraining Binary

RelationshipEffect of the Imposed Constraining Relationship on

the Embedded Relationship1:1:1 X:Y is M:1 X:Y is M:1

X:Z is M:1Y:Z is M:N

X:Y is 1:M X:Y is 1:MX:Z is M:NY:Z is M:1

X:Y is 1:1 X:Y is 1:1X:Z is M:1Y:Z is M:1

M:1:1 X:Y is M:1 X:Y is M:1X:Z is M:1Y:Z is M:N

X:Z is M:1 X:Y is M:1X:Z is M:1Y:Z is M:N

Y:Z is M:1 X:Y is M:NX:Z is M:NY:Z is M:1

Y:Z is 1:M X:Y is M:NX:Z is M:NY:Z is 1:M

Y:Z is 1:1 X:Y is M:NX:Z is M:NY:Z is 1:1

M:N:1 X:Z is M:1 X:Y is M:NX:Z is M:1Y:Z is M:N

Y:Z is M:1 X:Y is M:NX:Z is M:NY:Z is M:1

Table 4: Effects of single binary imposition on a ternary relationship [JONE96]

Figure 8 is another example ER diagram showing a ternary relationship without an imposed

constraining relationship. This diagram contains a ternary relationship and two cyclic paths,

XYZWVX and XZWVX, that requires evaluation. Both cyclic paths involve embedded binary

relationships. First, the ternary is evaluated for structural validity. Since in this example there is no

constraining relationships on the ternary then the ternary relationship is structurally valid. The analysis

of the cyclic paths for structurally validity is restricted only to the explicit or implicit binary

relationships in the paths. Both cyclic paths in Figure 8 contain at least one ‘Many to Many’ and

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therefore according to Rule 5 these paths are structurally valid. This diagram is structurally valid

because all the paths and the ternary relationship are structurally valid.

FIGURE 8: An example of a valid multi-path Entity Relationship diagram containing anunconstrained ternary relationship showing the embedded binary relationships

BUDGET(Y) IMPLEMENTS TEAM

(Z)

PROJECT(X)

M

11

CONTRACTWITH

M

M

CONSULTINGFIRM

(V)

CONSULTANTS(V)SUPPORTS

WORKSFOR

1

11

M

N

M

NM

N

Figure 9 is an example of an ER diagram containing a ternary relationship with an imposed

constraining relationship. We will use the notation R(xyz)|R(xy) to indicate that R(xy) is a

constraining relationship on R(xyz). This diagram contains the constrained ternary relationship

R(xyz)|R(xy) and two cyclic paths, XYZWVX and XZWVX. The first step again is to analyze the

ternary relationship with the constraining relationships imposed on it. R(xy) is a M:1 constraining

relationship and from Table 3 or the EBP rule we conclude it is a valid imposition on a M:1:1 ternary

relationship. Therefore the ternary is structurally valid. In our analysis of the cyclic paths we only

use the binary relationships. In Figure 9 both cyclic paths XYZWVX and XZWVX are invalid

according to our rules. They are called circular relationships [TILL93] [HOWE89] where the

relationships are all mandatory and ‘Many to One’ in only one direction. This ER diagram in Figure 9

is invalid and is the same diagram presented in Figure 1 of this paper.

BUDGET(Y) IMPLEMENTS TEAM

(Z)

M

11

CONTRACTWITH

M

M

CONSULTINGFIRM

(V)

CONSULTANTS(V)SUPPORTS

WORKSFOR

1

11

M

N

M

1M

1

FIGURE 9: An example of an invalid multi-path Entity Relationship diagram with a constrainedternary relionship showing the effects of the embedded relationships

FUNDED

M

1

PROJECT(X)

16

From the ideas presented in this section we conclude that first the ternary relationship must be

evaluated for its structural validity and the embedded binary relationships between the entities must be

determined. When that step is completed we can identify and evaluate all the paths for structural

validity. In evaluating the ternary relationship we use the following rule:

Rule 8: If the maximum cardinality constraints for a constraining binary relationship imposed

on a ternary relationship is greater than or equal to the maximum cardinality constraints of the

ternary relationship between the two involved entities then the constrained ternary relationship is

valid.

4.2.4 The Effects of Multiple Constraining Relationships on Structural Validity

In the previous section we showed that the imposition of a constraining relationship affects the

embedded binary relationships between the entities of the ternary relationship and consequently can

affect the validity of the ER diagram. If necessary an additional constraining relationship can be

imposed on an already constrained ternary relationship in order to further define the modeler’s

concept of the real world environment. Care must be taken when imposing an additional constraining

relationship on an already constrained relationship. The first constraining relationship imposes

additional derived functional dependencies on the functional dependencies already imposed by the

ternary relationship. A second constraining relationship can not redefine an already defined functional

dependencies, although a valid imposition may indirectly further define the derived embedded

cardinality between two entities. The only entities in a ternary relationship available to be constrained

are those that have an embedded ‘Many to Many’ cardinality constraint between them. These entities

do not have binary type functional dependencies between them and are therefore unconstrained with

respect to a binary relationship.

For example if in a 1:1:1 ternary relationship R(xyz) we choose any two entities, such as X

and Y, and impose a 1:1 constraining relationship between X and Y then we introduce two additional

functional dependencies on the ternary relationship.

Original FDs on R(xyz) Imposed FDs from R(xy)

(X,Y) → Z Y → X(X,Z) → Y X → Y(Y,Z) → X

17

Two additional FDs are derived: Y → Z and X → Z. From this information each embedded binary

relationship in the ternary relationship is defined by a functional dependency, therefore no additional

constraining relationships can be imposed.

In another example of a 1:1:1 ternary relationship R(xyz) we choose any two entities, such as

X and Y, and impose a M:1 constraining relationship between X and Y. This would impose one

binary functional dependency on the ternary and one additional FD is derived: X → Z.

Original FDs on R(xyz) Imposed FDs from R(xy) Derived FD

(X,Y) → Z X → Y X → Z(X,Z) → Y(Y,Z) → X

From this information only two embedded binary relationships in the ternary relationship are

constrained by binary functional dependencies. The embedded binary association between entities Y

and Z is still ‘Many to Many’ allowing an additional constraining relationship to be imposed if

necessary. We have three options to analyze. The relationship between Y and Z can be M:1, 1:M, or

1:1. The only other option of imposing M:N would be redundant. The following are the functional

dependencies for the three cases.

R(xyz) constrained by R(xy) Imposed R(yz) FDs from R(yz) Derived FD

(X,Y) → Z Many to One Y → Z Y → X(X,Z) → Y(Y,Z) → X One to Many Z → Y Z → XX → YX → Z (derived) One to One Y → Z Y → X

Z → Y Z → X

In the ‘Many to One’ case for R(yz) the derived FD {Y → X} in combination the FD {X → Y} from

the explicit previously imposed relationship is redefining the relationship R(xy) and therefore is

invalid. The same argument applies to the ‘One to One’ case. The only valid imposition is the ‘One

to Many’ case because this imposition does not redefine any explicitly defined relationship. It does

further define the embedded relationship R(xz) to be ‘One to One’. Table 5 from [JONE96] shows

the results of a second constraining binary imposition on ternary relationships for all possible relevant

cases.

18

TernaryRelationship

X:Y:Z

Imposed ConstrainingBinary Relationship

Embedded BinaryRelationship

Additional ImposedConstraining Binary

Relationship

Resultant EmbeddedBinary Relationship

1:1:1 X:Y is M:1 X:Y is M:1 Y:Z is 1:M X:Y is M:1X:Z is M:1 X:Z is 1:1Y:Z is M:N Y:Z is 1:M

X:Y is M:1 X:Y is M:1 X:Z is 1:M X:Y is M:1X:Z is M:1 X:Z is 1:1Y:Z is M:N Y:Z is 1:M

M:1:1 X:Y or Z:X is M:1 X:Y is M:1 Y:Z is 1:1 X:Y is M:1X:Z is M:1 X:Z is M:1Y:Z is M:N Y:Z is 1:1

X:Y or Z:X is M:1 X:Y is M:1 Y:Z is M:1 X:Y is M:1X:Z is M:1 X:Z is M:1Y:Z is M:N Y:Z is M:1

X:Y or Z:X is M:1 X:Y is M:1 Y:Z is 1:M X:Y is M:1X:Z is M:1 X:Z is M:1Y:Z is M:N Y:Z is 1:M

Y:Z is 1:1 X:Y is M:N X:Y or X:Z is M:1 X:Y is M:1X:Z is M:N X:Z is M:1Y:Z is 1:1 Y:Z is 1:1

Y:Z is M:1 X:Y is M:N X:Y or X:Z is M:1 X:Y is M:1X:Z is M:N X:Z is M:1Y:Z is M:1 Y:Z is M:1

Y:Z is 1:M X:Y is M:N X:Y or X:Z is M:1 X:Y is M:1X:Z is M:N X:Z is M:1Y:Z is 1:M Y:Z is 1:M

M:N:1 X:Z is M:1 X:Y is M:N Y:Z is M:1 X:Y is M:NX:Z is M:1 X:Z is M:1Y:Z is M:N Y:Z is M:1

Y:Z is M:1 X:Y is M:N X:Z is M:1 X:Y is M:NX:Z is M:N X:Z is M:1Y:Z is M:1 Y:Z is M:1

Table 5: The effects of multiple binary imposition on ternary relationships [JONE96]

From the ideas presented in this section we can state Rule 9 below.

Rule 9: If a second constraining binary relationship is required to further define an already

constrained ternary relationship then it can only be imposed between two entities where the

maximum cardinality constraint is ‘Many to Many’ and the effect of the second constraining

relationship can not redefine any previously defined explicit relationships or relax any

previously derived binary relationships for the imposition to be structurally valid.

4.2.5 The Effect of Minimum Cardinality Constraints of Implicit and Explicit BinaryRelationships on Structural Validity

Implicit Binary Relationships

The previous section did not mention minimum cardinality constraints. The purpose of this

section is to explore the effects of minimum cardinality on implicit relationships and explicit

constraining relationships. Imposing an optional participation constraint on one (or more) of the

entities in a ternary relationship restricts at least one instance of the entity from participating in the

19

ternary relationship. We will continue to use instance tables as an analysis methodology and

generalize from the results. In Figure 10 we use four triples to reveal the ‘Many to Many’

embedded binary relationships in the ternary. We use instance Z9 in entity Z to represent the set

of instances that do not participate in the ternary relationship. We extract the embedded binary

relationships R(xz), R(yz), and R(xy) from the instance triples of the ternary relationship R(xyz).

Since Z9 did not participate in R(xyz) it follows that is does not participate in R(xz) and R(yz).

We conclude that when evaluating structural validity the minimum cardinality constraints of the

embedded binary relationships must follow the minimum cardinality constraints on the entities of

the ternary relationship regardless of the ternary’s maximum cardinality. The driving reason is

that the instance pairs in the embedded relationships must be a subset of the of the instance triples

of the ternary relationship.

Figure 10: A 1:1:1 ternary relationship with optional participation on Z and related instance tables(Z9 does not participate)

X R(xyz) Y

Z

1 1

1

N

M

M

M

N

N

R(xyz)

X1:Y1:Z1X1:Y2:Z2X2:Y1:Z3X2:Y2:Z1

R(xz)

X1:Z1X1:Z2X2:Z3X2:Z1

R(yz)

Y1:Z1Y2:Z2Y1:Z3Y2:Z1

Instance Tables

R(xy)

X1:Y1X1:Y2X2:Y1X2:Y2

X Y Z

X1 Y1 Z1X2 Y2 Z2 Z3 Z9

Z9 does not participate in R(xyz)

Explicit Binary Relationships

In this section we examine minimum cardinality constraints with respect to explicit binary constraining

relationships. Again we will use the concept of minimally with instance tables and functional

dependencies to perform our analysis. In our tables let Z9 represent the set of instances that do not

participate in the relationship. We will first examine the imposition of a constraining relationship on a

ternary relationship where the minimum cardinality of the binary follows the minimum cardinality of

the ternary. Consider Figure 11 with an imposition of a ‘Many to One’ binary constraining

relationship between entities Y and Z, and the following possible scenario. This imposition adds the

functional dependency Z → Y to the relationship removing Y2:Z1 from instance table R(yz) and

X2:Y2:Z1 from R(xyz) as shown in Figure 11. We also lose X2:Z1 from R(xz) because of the

change in the embedded cardinality between Z and X caused by the constraining relationship (See

Table 4). The remaining instance groupings in table R(xyz)|R(yz) still contains the minimal number of

20

triples (X1:Y1:Z1) to represent the ternary relationship and R(xy) remains a ‘Many to Many’

embedded relationship. In the instance table in figure 11 R(xz) and R(yz) correctly reflect ‘Many to

One’ relationships as shown in Table 4. Figure 11 and the accompanying instance tables shows that

the effect of the constraining relationship R(yz) has on R(xyz). If we remove Z9 from the instance

table and imposed mandatory participation on Z from both the ternary and the constraining binary we

find the results to be the same as in the instance tables of Figure 11. We conclude that when the

minimum cardinality constraint of the constraining binary relationship follows the minimum cardinality

of ternary the resulting embedded binary relationships remain to be subsets of both the original and

the constrained ternary relationships.

Figure 11: A 1:1:1 ternary relationship with optional participation on Z (Z9 does not participate)and a constraining relationship R(yz) imposed on R(xyz)

R(xyz)|R(yz)

X1:Y1:Z1X1:Y2:Z2X2:Y1:Z3

R(xyz)

X1:Y1:Z1X1:Y2:Z2X2:Y1:Z3X2:Y2:Z1

R(yz)

Y1:Z1Y2:Z2Y1:Z3

Instance Tables

X Y Z

X1 Y1 Z1X2 Y2 Z2 Z3 Z9

X R(xyz) Y

Z

1 1

1 1

M

1

M

N

1

M

R(yz)M

R(xz)

X1:Z1X1:Z2X2:Z3

R(xy)

X1:Y1X1:Y2X2:Y1

We further find that from our definition of a constraining relationship the instance pairs of

R(yz) are always a subset of the instance triples of R(xyz). If R(yz) contains an instance pair that is

not a subset of R(xyz) then the binary relationship is unrelated to the ternary relationship and not a

constraining relationship. Consider the case where the minimum cardinality constraint is ‘optional’ on

one of entities of the ternary relationship that is being constrained. In Figure 12 the proposed

constraining relationship R(yz) is ‘mandatory’ forcing Z9 to participate in R(yz) without participating

in R(xyz). Y1:Z9 in R(yz) is not a member of R(xyz) therefore R(yz) is not a subset of the R(xyz),

therefore R(yz) can not be a constraining relationship. If the intent is for that relationship to be

constraining then it is semantically and structurally invalid because it is unrelated to the ternary

relationship. Figure 12 shows this example of an invalid constraining relationship R(yz).

21

Figure 12: A 1:1:1 ternary relationship with optional participation on Z (Z9 does not participate)and an invalid constraining relationship R(yz) with mandatory participation

R(xyz)

X1:Y1:Z1X1:Y2:Z2X2:Y1:Z3X2:Y2:Z1

R(yz)

Y1:Z1Y2:Z2Y1:Z3

Y1:Z9

Instance Tables

X Y Z

X1 Y1 Z1X2 Y2 Z2 Z3 Z9

X R(xyz) Y

Z

1 1

1 1

M

1

M

N

1

M

R(yz)M

R(yz) is not constrainingbecause the instancesof R(yz) is not a subsetof the instances ofR(xyz)

The case where the ternary is fully mandatory and the proposed constraining relationship is

‘optional’ on one side presents a similar issue. The driving force in the resulting constrained

relationship is the ternary relationship not the constraining relationship. In this case the ternary

relationship is allowing all instances to participate in the relationship. The constraining binary

relationship is in conflict with the ternary relationship in that it restricts at least one instance that is not

restricted by the ternary. In this case the constraining relationship is redefining the ternary

relationship not further defining it.

Rule 10: The minimum cardinality constraints of a constraining relationship must follow the

minimum cardinality constraints of the ternary relationship being constrained for it to be

structurally valid.

5. Conclusion

In this paper we summarized the rules regarding structural validity of binary relationships in

entity-relationship diagramming from a previous study. We capitalized on these rules to perform a

complete investigation of structural validity of ternary relationships. We distinguished between

structural validity and semantic validity in that the structured macro view of the diagram supports the

semantic micro view of the data model. The analysis of semantic validity requires a more subjective

look at actual real world requirements of the user community as mapped to subsections of the model

while the analysis of the structural validity can be performed in a pragmatic approach. In previous

approaches only maximum cardinality constraints were used in their analysis. Here we use both

maximum and minimum cardinality constraints yielding a more complete analysis of the structural

validity of ternary relationship types. Our study also includes an analysis of ternary relationships and

unrelated binary relationships. Previous studies only performed standalone validity evaluations of

ternary relationships. In our study we consider the ternary relationship as part of the overall diagram

and our rules address ternary relationships as they coexist with other relationships in a path structure

22

within the model. Our analysis extended the original six rules to ten, and presented generic examples

along with the theoretical foundations for each new rule. We presented a comprehensive discussion

of constraining binary relationships and their use to more precisely define the ternary relationship to

model more complex business concepts. We addressed both the issues of minimum and maximum

cardinality constraints in the evaluation of the structural validity of ternary relationships. We found

that when the minimum cardinality of any constraining relationship follows the minimum cardinality of

the ternary relationship the validity of the constrained relationship is determined by the EBP rule, but

when the minimum cardinalities are different the constrained relationships are invalid. The rules we

present are easy to use in the evaluation of the structural validity of complex entity-relationship

diagrams containing binary and ternary relationships. The ten rules add value to the analysis and

design process insofar as they provide a standalone, application-independent tool that can easily be

automated to evaluate the structural validity of an entity relationship diagram. We summarize our

rules in Appendix One.

Additional research in this area would allow an extension of these rules into other data

modeling techniques and the discipline of object modeling. We believe that this analysis is a

completed effort and can be readily implemented in its current form providing an adequate foundation

for the evaluation of structural validity in entity relationship modeling. These rules can be extended

to the analysis of other diagramming techniques used in data modeling, and the analysis of diagrams in

the object-oriented model.

23

References

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[BOEH76] Boehm, Barry W., 1976. Software Engineering, IEEE Transactions on Computers, 25(12)1226-1241,December 1976.

[BOEH81] Boehm, Barry W., 1981. Software Engineering Economics, Prentice-Hall, Inc. Englewood Cliffs, NJ

[CHEN76] Chen, Peter, 1976. “The Entity-Relationship Model -- Toward a Unified View of Data”, ACMTransactions on Database Systems, 1(1)9-36, March 1976.

[CODD70] Codd E. F., 1970. A Relational Mode of Data for Large Shared Data Banks, Communications of the ACM,13(6)377-387, June, 1970.

[DALY77] Daly, E. B., 1977. Management of Software Engineering, IEEE Transactions on Software Engineering,3(3)229-242, May, 1977

[DELO73] Delobel, C. and R. C. Casey, 1973. Decomposition of a Data Base and the Theory of Boolean SwitchingFunctions, IBM Journal of Research Development, 21(5)484-485, September 1973.

[DULL97] Dullea, James and Il-Yeol Song, 1997. “An Analysis of Cardinality Constraints in RedundantRelationships”, Accepted for publication in The Proceedings of the Sixth International Conference on Information andKnowledge Management, Las Vegas, Nevada, November 10-14, 1997.

[DULL98] Dullea, James and Il-Yeol Song, 1998. “An Analysis of Structural Validity in Unary and BinaryRelationships in Entity Relationship Modeling”, Currently submitted for publication.

[ELMA94] Elmasri, Ramez and Shamkant B. Navathe, 1994. Fundamentals of Database Systems, 2nd Ed., TheBenjamin/Cummings Publishing Co, Inc., Redwood City, CA.

[FAGA74] Fagan, M. Design and Code Inspections and Process Control in the Development of Programs, IBM ReportIBM-SDD-TR-21-572, December, 1974.

[HOWE89] Howe, D. R., 1989. Data Analysis for Data Base Design, 2nd Ed., Edward Arnold, London, GB.

[JONE96] Jones, Trevor H., and Il-Yeol Song, 1996. “Analysis of Binary/Ternary Cardinality Combinations in Entity-Relationship Modeling”, Data & Knowledge Engineering, 19(1996)39-64.

[SONG93] Song, Il-Yeol and Trevor H. Jones, 1993. “An Analysis of Binary Relationships within TernaryRelationships in ER Modeling”, Proceedings of the 12th International Conference on Entity-Relationship Approach, pp265-276, Arlington, TX December 15-17, 1993.

[SONG95] Song, Il-Yeol, Mary Evans, and E. K. Park, 1995. “A Comparative Analysis of Entity-RelationshipDiagrams”, Journal of Computer & Software Engineering, 3(4)427-459.

[TEOR94] Teorey, Toby J., 1994. Database Modeling and Design - The Entity-Relationship Approach, MorganKaufmann Publishers, Inc., San Mateo, CA.

[TILL93] Tillman, George 1993. A Practical Guide to Logical Data Modeling, McGraw-Hill, Inc., New York

24

Appendix 1

A summary of the rules we developed regarding binary and ternary relationships is as follows:

1. An acyclic path containing binary relationships or embedded binary relationships from ternary relationships is

always structurally valid.

2. A cyclic path containing all ‘One-to-One’ Mandatory-Mandatory binary relationships (explicit or embedded) is

always structurally valid.

3. A cyclic path that contains all binary relationships (explicit or embedded), and one or more ‘Optional-Optional’

relationships is always structurally valid.

4. A cyclic path that contains all binary relationships (explicit or embedded), and one or more ‘Many-to-One’

relationships with ‘Optional’ participation on the ‘One’ side is always structurally valid.

5. A cyclic path that contains all binary relationships (explicit or embedded), and one or more ‘Many-to-Many’

relationships is always structurally valid.

6. Cyclic paths containing at least one binary relationship (explicit or embedded) that is [{‘One-to-Many’, Totally

Mandatory}, {‘One-to-Many’, Mandatory-Optional}, or {‘One-to-One’, Mandatory-Optional}] and at least one

other binary relationship that is [{‘Many-to-One’, Totally Mandatory}, {‘Many-to-One’, Optional-Mandatory },

or {‘One-to-One’, Optional-Mandatory}] is structurally valid.

7. If a cyclic path contains a ternary relationship where there is no explicit restricting binary relationships on the

entities, then the path is always structurally validity, regardless of maximum and minimum cardinality constraints

on the ternary relationship

8. If the maximum cardinality constraints for a constraining binary relationship imposed on a ternary relationship is

greater than or equal to the maximum cardinality constraints of the ternary relationship between the two involved

entities then the constrained ternary relationship is valid.

9. If a second constraining binary relationship is required to further define an already constrained ternary

relationship then it can only be imposed between two entities where the maximum cardinality constraint is ‘Many

to Many’ and the effect of the second constraining relationship can not redefine any previously defined explicit

relationships or relax any previously derived binary relationships for the imposition to be structurally valid

10. The minimum cardinality constraints of a constraining relationship must follow the minimum cardinality

constraints of the ternary relationship being constrained.