Preface

Author Ondřej Pavlata Jablonec nad Nisou Czech Republic

Document date Initial release December 12, 2012 Last major release December 12, 2012 Last update March 12, 2015 (see Document history)

Warning

1. This document has been created without any prepublication review except those made by the author himself.

Table of contents

Overview

Subsequently, we define a generalization of the S1_ϵ structure, the canonical structure (O, ϵ) of object membership. (∗) The generalized structure is denoted S1_o and called the ontological structure of object membership. We then show that the restriction to primary objects, S1_o.pr, is a specialization of the core structure of RDF Schema. Moreover, the additional constraints that are not guaranteed in RDF Schema, can be considered to be satisfied by most reasonable RDFS structures. In particular, the built-in structures of RDF Schema and OWL 2 satisfy all the axioms of S1_o.pr structures.

Note:(∗) For simplicity, we assume condition (S1_ϵ~5⁺) so that S1_o is just an eigenclass completion of the primary structure S1_o.pr.

Preliminaries

Relational indiscernibility

• (1) a R x iff b R x,   i.e. a and b have have equal images under R ∖ X.∆.
• (2) x R a iff x R b   i.e. a and b have have equal (preimages)
• (3) a R a iff b R b.

Given a system (X, R) of relations on a set X (so that R ⊆ ℙ(X × X)) elements a and b of X are indiscernible w.r.t. R if they are indiscernible w.r.t. R for each R ∈ R.

For a relational system (X, R) we distinguish the following finiteness conditions:

Note: A subset Y of X is said to be cofinite if X ∖ Y is finite.

Proposition:

1. For a relational system (X, R), the following are equivalent:
   • (IND~F~1)
   • R is finite and every R has only finitely many blocks of indiscernibility.
2. If (X, R) satisfies (IND~F~2) (and in particular, if it satisfies (IND~F~3)) then for every relation R ∈ R, acyclic paths in R are of limited length, i.e. there exists n > 0 such that every path
   • x₀ R x₁ R x₂ … x_n-1 R x_n
   has a cycle (i.e. x_i = x_j for some i ≠ j).
3. Conditions (IND~F~3), (IND~F~3') and (IND~F~3'') are equivalent.
4. If (IND~F~3') is satisfied and X is infinite then there is exactly one cofinite block of indiscernibility.

RDFS structures

1. RDFS₁ structures capture all the information related to object membership.
2. RDFS₂ structures contain the built-in RDFS structure (with all axiomatic triples).
3. RDFS₃ structures distinguishes literals in their embedding into blank nodes.

• Typing of literals.
• Datatype entailment.

RDFS₁: The core structure of RDF Schema

• Ō is set of objects.
• Ψ is a subset of Ō × Ō × Ō (= Ō³).
• The remaining 8 symbols denote distinguished objects with the following terminology:

  Object	RDFS name	Terminology
  ‹ϵ›	rdf:type	instance-of   (inverse: (a-)type-of, (a-)class-of)
  ‹≤C›	rdfs:subClassOf	subclass-of   (inverse: superclass-of)
  ‹≤P›	rdfs:subPropertyOf	subproperty-of   (inverse: superproperty-of)
  ‹ϵd›	rdfs:domain	domain-restricted-by   (inverse: domain-restrictor-of)
  ‹ϵr›	rdfs:range	range-restricted-by   (inverse: range-restrictor-of)
  r	rdfs:Resource	resource classifier (inheritance root)
  c	rdfs:Class	class classifier (metaclass root / instance root)
  p	rdf:Property	property classifier

• Elements of Ō³ are triples of objects. For a triple t = (s,p,o),
  • s is the subject of t,
  • p is the predicate of t,
  • o is the object of t.
  We denote .s, .p, and .o the respective projections from Ō³ to Ō.
• We also denote .so the projection from Ō³ to Ō² defined by (s,p,o).so = (s,o).
• The relational extent of an object p with respect to a triples set X ⊆ Ō³, is denoted p.rel(X) and defined by
  • p.rel(X) = {(s,o) | (s,p,o) ∈ X}.
• We will be mostly concerned with triples from Ψ, so that we abbreviate p.rel = p.rel(Ψ), i.e.
  • p.rel = {(s,o) | (s,p,o) ∈ Ψ}.
• Using the image notation, Ō.rel denotes the relational system of Ψ.
• For a triple t = (s,p,o) we denote t.so = (s,o). Therefore, Ψ.so denotes the projection of Ψ to the corresponding subject-object pairs.
• For each of the 5 symbols of the form ‹s› denoting a distinguished object x we introduce the relational symbol s that denotes the relation x.rel. For example, ϵ is a relation between objects such that
  • x ϵ y   iff   (x,y) ∈ ‹ϵ›.rel     (that is, iff (x,‹ϵ›,y) ∈ Ψ).
• Terminology introduced for ‹s›-symbols applies to their respective s-symbols. For example, ϵ is the instance-of relation. If x ϵ y then x is an instance of y.
• For an object x, we write x.϶ / x.ϵ for the preimage / image of {x} under ϵ. In particular x.϶ is the set of all instances of x.
• We use similar notation for the ϵ_d and ϵ_r relations. For an object x, elements of the set x.ϵ_d (resp. x.ϵ_r) are domain-restrictors (resp. range-restrictors) of x.
• We write x.↧_C / x.↥_C for the preimage / image of {x} under ≤_C. Similar notation is applied for sets of objects.
• Resources are elements of r.϶, i.e. instances of r.
• Classes are elements of c.϶, i.e. instances of c.
• Properties are elements of p.϶, i.e. instances of p.
• Helix classes are elements of c.↥_C, i.e. superclasses of c.
• Metaclasses are elements of c.↧_C, i.e. subclasses of c.

The structure is subject to the following axioms:

• (RDFS₁~1)
  
  • (a) Ō = Ψ.s ∪ Ψ.p ∪ Ψ.o, i.e. every object appears in Ψ.
  • (b) The 8 distinguished objects are pairwise distinct.
• (RDFS₁~2)
  The following 10 triples are in Ψ (they are axiomatic triples []):

  Axiom	Consequence entailed by rdfs2 / rdfs3	Description
  ‹ϵ› ϵd r	(ϵ) ⊆ (r.϶) × (c.϶)	Only classes can have instances.
  ‹ϵ› ϵr c
  ‹≤C› ϵd c	(≤C) ⊆ (c.϶) × (c.϶)	The subclass-of relation is only defined between classes.
  ‹≤C› ϵr c
  ‹≤P› ϵd p	(≤P) ⊆ (p.϶) × (p.϶)	The subproperty-of relation is only defined between properties.
  ‹≤P› ϵr p
  ‹ϵd› ϵd p	(ϵd) ⊆ (p.϶) × (c.϶)	Only properties can have domain-restrictors.
  ‹ϵd› ϵr c		Only classes can be domain-restrictors.
  ‹ϵr› ϵd p	(ϵr) ⊆ (p.϶) × (c.϶)	Only properties can have range-restrictors.
  ‹ϵr› ϵr c		Only classes can be range-restrictors.

• (RDFS₁~3) The following RDFS rules [] apply:

  Rule name		Convenient expression	Description	Triple inference
  RDFS	OWL 2
  rdfs2	prp-dom	x ϵd y implies dom(x.rel) ⊆ y.϶		x ϵd y, a x ?	→	a ϵ y
  rdfs3	prp-rng	x ϵr y implies rng(x.rel) ⊆ y.϶		x ϵr y, ? x a	→	a ϵ y
  rdfs4a		Ψ.s ⊆ r.϶	The subject of every triple is a resource.	x ? ?	→	x ϵ r
  rdf1		Ψ.p ⊆ p.϶	The predicate of every triple is a property.	? x ?	→	x ϵ p
  rdfs4b		Ψ.o ⊆ r.϶	The object of every triple is a resource.	? ? x	→	x ϵ r
  rdfs7	prp-spo1	x ≤P y implies x.rel ⊆ y.rel		x ≤P y, a x b	→	a y b
  rdfs8		c.϶ ⊆ r.↧C	All classes are subclasses of r.	x ϵ c	→	x ≤C r
  rdfs9	cax-sco	(ϵ) ○ (≤C) ⊆ (ϵ)	Instances of a class y are (among) instances of every superclass of y.	x ϵ y, y ≤C z	→	x ϵ z
  rdfs10		≤C is reflexive on c.϶	The subclass-of relation, ≤C, is a preorder on classes.	x ϵ c	→	x ≤C x
  rdfs11	scm-sco	≤C is transitive		x ≤C y, y ≤C z	→	x ≤C z
  rdfs5	scm-spo	≤P is transitive	The subproperty-of relation, ≤P, is a preorder on properties.	x ≤P y, y ≤P z	→	x ≤P z
  rdfs6		≤P is reflexive on p.϶		x ϵ p	→	x ≤P x

• (RDFS₁~4) The following extended RDFS rules [] apply:

  Rule name		Convenient expression	Description	Triple inference
  RDFS	OWL 2
  ext1	scm-dom1	(ϵd) ○ (≤C) ⊆ (ϵd)	Superclasses of domain-restrictors of x are domain-restrictors of x.	x ϵd y, y ≤C z	→	x ϵd z
  ext2	scm-rng1	(ϵr) ○ (≤C) ⊆ (ϵr)	Superclasses of range-restrictors of x are range-restrictors of x.	x ϵr y, y ≤C z	→	x ϵr z
  ext3	scm-dom2	(≤P) ○ (ϵd) ⊆ (ϵd)	Domain-restrictors of superproperties of x are domain-restrictors of x.	x ≤P y, y ϵd z	→	x ϵd z
  ext4	scm-rng2	(≤P) ○ (ϵr) ⊆ (ϵr)	Range-restrictors of superproperties of x are range-restrictors of x.	x ≤P y, y ϵr z	→	x ϵr z
  ext5		‹ϵ›.ϵd ⊆ r.↥C	Domain-restrictors of ‹ϵ› are equivalent to r.	‹ϵ› ϵd x	→	r ≤C x
  ext6		‹≤C›.ϵd ⊆ c.↥C	Domain-restrictors of ‹≤C› are superclasses of c.	‹≤C› ϵd x	→	c ≤C x
  ext7		‹≤P›.ϵd ⊆ p.↥C	Domain-restrictors of ‹≤P› are superclasses of p.	‹≤P› ϵd x	→	p ≤C x
  ext8		‹≤C›.ϵr ⊆ c.↥C	Range-restrictors of ‹≤C› are superclasses of c.	‹≤C› ϵr x	→	c ≤C x
  ext9		‹≤P›.ϵr ⊆ p.↥C	Range-restrictors of ‹≤P› are superclasses of p.	‹≤P› ϵr x	→	p ≤C x

• (RDFS₁~5) The finite discernibility condition (IND~F~3) applies:
  • The relational system Ō.rel has a cofinite block of indiscernibility.
  As a consequence, acyclic chains in ϵ (or in ≤) are of limited length.

Notes:

• Boundedness of acyclic chains in ϵ can also be ensured by finiteness of c.϶ ∖ p.϶, i.e.
  • there are only finitely many classes that are not properties.
  This condition is valid in RDFS₂ structures (and thus in OWL2₁ structures). However, we prefer to use the notion of indiscernibility due to the descriptivness it provides.
• In the specification [], rules ext5–ext9 are expressed with the composition (≤_P) ○ (ϵ_d) (resp. (≤_P) ○ (ϵ_r)) instead of just ϵ_d (resp. ϵ_r). However, due ext3 and ext4 (and by reflexivity of ≤_P), the composition with ≤_P is redundant.

Proposition:

1. p ϵ c.
2. Every object is a resource, Ō = r.϶.
3. In rdfs8, rdfs9, and ext1–ext9, set inclusion, ⊆, can be equivalently replaced by equality, =. In particular:
   • (ϵ) ○ (≤_C) = (ϵ).
   • (≤_P) ○ (ϵ_d) ○ (≤_C) = (ϵ_d).
   • (≤_P) ○ (ϵ_r) ○ (≤_C) = (ϵ_r).

Proof:

1. Since Ψ is non-empty we have ∅ ≠ Ψ.p ⊆ p.ϵ (by rdf1). Thus p ∈ rng(ϵ). Using ‹ϵ› ϵ_r c and rdfs3, we have rng(ϵ) ⊆ c.϶, thus p ϵ c.
2. If x ∈ Ψ.s (resp. x ∈ Ψ.o) then x ϵ r by rdfs4a (resp. rdfs4b). If x ∈ Ψ.p then x ϵ p by rdf1. Since p is a class due to the previous proposition, it is also a subclass of r (by rdfs8). We have
   • x ϵ p ≤_C r.
   By rdfs9, x ϵ r.

RDFS₁ closure

• All axiomatic triples are in X.
• All entailment rules from (RDFS₁~3) and (RDFS₁~4) are satisfied for X, i.e. triples inferred from triples from X are in X.

Proposition:

1. Λ₁-closed sets of triples form a closure system in (ℙ(Ō³), ⊆). We use the Λ₁ symbol for the corresponding closure operator, i.e. Λ₁(T) is the smallest (w.r.t. set inclusion) superset of T that is Λ₁-closed.
2. Λ₁ preserves the finiteness condition (RDFS₁~5), i.e. given a set T of triples, if is a cofinite set X of objects that are mutually indiscernible w.r.t. T, then for some cofinite subset Y of X, all objects of Y are indiscernible w.r.t. Λ₁(T).
3. Corollary: Adding finitely many triples to Ψ and subsequently applying Λ₁-closure results in a (new) RDFS₁ structure.

Proof:

2. Let Y be the (necessarily cofinite) subset of X of objects that do not appear as a constant in a triple pattern. We show that all elements of Y are mutually indiscernible w.r.t. Λ₁(T).

   Let a, b be from Y, p a property and o an objects different from both a and b. We should show that

   • (a,p,o) ∈ Λ₁(T) iff (b,p,o) ∈ Λ₁(T).
   • (o,p,a) ∈ Λ₁(T) iff (o,p,b) ∈ Λ₁(T).
   • (a,p,a) ∈ Λ₁(T) iff (b,p,b) ∈ Λ₁(T).
   But this follows, since the entailment rules do not contain a triple pattern of the form (x,x,a) or (a,x,x) (or even (x,x,x)) where x is a variable.

The minimum RDFS₁ structure

• ϵ is obtained as (≤_C) ○ A ○ (≤_C).   (A = p.϶ × {p} ∪ {(r,c)}.)
• ≤_C is obtained as the reflexive transitive closure of (c.϶, C).   (C = {p, c} × {r}.)
• ≤_P is obtained as the reflexive transitive closure of (p.϶, P).   (P = ∅.)
• ϵ_d is obtained as (≤_P) ○ D ○ (≤_C).   (D = {‹≤_P›, ‹ϵ_d›, ‹ϵ_r›} × {p} ∪ {(‹ϵ›,r), (‹≤_C›,c)}.)
• ϵ_r is obtained as (≤_P) ○ R ○ (≤_C).   (R = {‹≤_C›, ‹ϵ_d›, ‹ϵ_r›} × {c} ∪ {(‹ϵ›,c), (‹≤_P›,p)}.)

RDFS₂: A superstructure of the built-in RDFS structure

• (RDFS₂~1) The structure has a built-in substructure shown by the diagram below.
• (RDFS₂~2) Additional entailment rules [] are applied:

  Rule name	Convenient expression	Description	Triple inference
  rdfs12	CMP.϶ ⊆ m.↧P	Instances of rdfs:ContainerMembershipProperty are subproperties of rdfs:member.	x ϵ CMP	→	x ≤P m
  rdfs13	D.϶ ⊆ L.↧C	Instances of rdfs:Datatype are subclasses of rdfs:Literal.	x ϵ D	→	x ≤C L

• 13 classes, including the 3 classes from the RDFS₁ structure (r, c, p).
• 16 properties other than instances of rdfs:ContainerMembershipProperty (CMP), including the 5 properties from the RDFS₁ structure,
• infinitely many properties rdf:_1, rdf:_2, …, which are all instances of CMP,
• 1 terminal object, rdf:nil, which is an instance of rdf:List.

Notes:

1. Missing arrows for ϵ_d / ϵ_r indicate that the only respective domain or range restrictor is the inheritance root, r.
2. It is correct to state that the built-in structure is the minimum RDFS₂ structure since the following are satisfied:
   • The structure is an RDFS₁ structure. In particular, the finiteness condition is satisfied: the infinitely many properties rdf:_1, rdf:_2, …, form a cofinite block of indiscernibility. (Moreover the properties do not appear as objects of non-loop triples.)
   • Condition (RDFS₂~2) is satisfied.

RDFS₂ closure

• All triples from the built-in RDFS₂ structure are in X.
• All entailment rules from (RDFS₁~3), (RDFS₁~4) and (RDFS₂~2) are satisfied for X.

Proposition:

1. Any Λ₂-closed set of triples is Λ₁-closed.
2. Λ₂-closed sets of triples form a closure system; let Λ₂ be the corresponding closure operator.
3. Λ₂ preserves finite discernibility (IND~F~3).
4. Λ₂ and Λ₁ preserve the following finiteness condition:
   • If the range of X.so ∖ Ō.∆ is finite then so is the range of Λ₂(X).so ∖ Ō.∆.
5. Corollary: For every RDFS₂ structure generated by a finite set of triples (via Λ₂) the range of Ψ.so ∖ Ō.∆ is finite, i.e. there are only finitely many non-leaf objects o such that
   • (s,p,o) ∈ Ψ for some objects s, p such that s ≠ o.

Proof:

4. For every entailment rule T → t = (s,p,o), either of the following is satisfied:
   • (a) o ∈ T.o.
   • (b) o ∈ {r, p}.
   • (c) s = o.
   If the rule is applied to a set X of triples, then in case (a) and (b) the range of X.so does not increase (is the same as the range of (X ∪ {t}).so). In the (c) case, t is a loop triple, (o,p,o), which is excluded from the finiteness constraint.

RDFS₃: Literals and blank nodes

• B is a set of objects called blank nodes or bnodes.
• L.b is a set of objects called literal bnodes.

Notes:

1. The specification of RDF Schema [] introduces two dual sets of entities:
   • The set L of literals.
   • The set L.b of their blank node counterparts.
   There is one-to-one mapping, .b, between literals and their counterparts, which assigns each literal x an object, x.b, called the blank node allocated to x. Literals are disjoint with objects, L ∩ Ō = ∅. Together with objects, they form the set of nodes. Objects that are not blank nodes are URI names.

   The set Ψ, as specified in [] (where it is called RDF graph), is allowed to also contain literals as the triple's object, i.e.

   • Ψ ⊆ Ō × (Ō ∖ B) × (Ō ⊎ L).
   The literal generalization (lg) and literal instantiation (gl) rules [] assert that for every objects a, b and every literal x,
   • (a,b,x) ∈ Ψ   iff   (a,b,x.b) ∈ Ψ.
2. The RDF Schema specification does not introduce any restrictions as to the range of the core 5 properties with respect to literals. For example,
   • ex:x rdf:type "5"   (x ϵ "5", x is an instance of "5")
   is a valid RDF triple. (In contrast, "5" rdf:type rdfs:Literal is not a valid RDF triple [].)

OWL-2 structures

OWL2₁: A superstructure of the built-in OWL 2 structure

• (OWL2₁~1) The structure has a built-in substructure shown by the diagram below.
• (OWL2₁~2) Additional entailment rules [] are applied:

  Rule name	Convenient expression	Description	Triple inference
  scm-cls	c.϶ ⊆ N.↥C	Every class is an ancestor of the bottom class owl:Nothing.	x ϵ c	→	N ≤C x

• (OWL2₁~3) The N class (owl:Nothing) has no instances, N.϶ = ∅.

• 58 classes:
  • 26 classes whose name starts with owl: followed by an upper case letter.
  • 2 classes with an owl: prefix followed by a lower case letter: owl:rational, owl:real. (∗)
  • 1 class prefixed rdf: – rdf:PlainLiteral.
  • 29 classes prefixed xsd:, with one distinguished class xsd:nonNegativeInteger (which is the smallest range-restrictor of 6 cardinality restriction properties). (∗)
  The latter 2 + 1 + 29 = 32 classes are datatypes – instances of rdfs:Datatype. There are 34 built-in datatypes in total. The remaining 2 classes from the RDFS₂ structure are rdfs:Literal and rdf:XMLLiteral. (However, the rdfs:Literal class is made an instance of rdfs:Datatype by an additional axiomatic triple, see notes below).
• 62 properties:
  • 51 properties with owl: prefix.
  • 11 facet names – rdf:langRange and 10 xsd: prefixed names.

• The (unified) inheritance, ≤, (equal to Ō.∆ ∪ (≤_C) ∪ (≤_P)) is the reflexive transitve closure of green lines (directed upwards whenever no arrow head is indicated).
• ≤_P is the restriction of ≤ to properties (in green).
• ≤_C is the restriction of ≤ to classes (which are in blue, brown or orange). In contrast to the built-in RDFS₂ structure, ≤_C is not antisymmetric. There are 4 equivalent pairs:
  • rdfs:Resource ↔ owl:Thing.
  • rdfs:Class ↔ owl:Class.
  • rdfs:Datatype ↔ owl:DataRange.
  • rdf:Property ↔ owl:ObjectProperty.
• ϵ is obtained as (≤) ○ A ○ (≤), where A is given by dashed dark blue arrows plus all imaginary arrows from a property to p.
• ϵ_d (resp. ϵ_r) is obtained as (≤_P) ○ R ○ (≤_C) where R is given by brown arrows whose tail is on the left side (resp.right side) of the source property plus all imaginary arrows from a property to r. (Therefore, rdfs:Resource and owl:Class are domain and range restrictors of every property.)

Notes and observations:

1. (∗) The set of built-in classes generated by the RDFS and OWL 2 RL generator service [] differs as follows:
   • The owl:real datatype is not present in [].
   • On the other hand, there are 2 additional datatypes: xsd:date and xsd:time.
2. Property groups indicated by gray dashed lines are blocks of indiscernibility in the built-in structure. The remaining non-grouped properties are singletons in w.r.t. indiscernibility, except for v, sA, dF, aP, aS, aT, tOP, and bOP which are all mutually indiscernible (they are characterized by not having ingoing or outgoing arrows in the diagram).
3. The two class groups indicated by green dashed lines are each a block of indiscernibility up to one missing class:
   • For datatypes, XL should also be grouped.
   • The DeP class is also indiscernible from TP.
4. Classes owl:AnnotationProperty and owl:OntologyProperty are minimal in inheritance but have 3 common instances, which therefore do not have least type.
5. There are changes in the least type of some objects from RDFS₂:

   Object x	The least type of x
   	In RDFS2	In OWL21
   rdfs:Literal	rdfs:Class	rdfs:Datatype
   rdfs:label, rdfs:comment, rdfs:seeAlso, rdfs:isDefinedBy	rdf:Property	owl:AnnotationProperty

6. After identification of equivalent classes, inheritance between built-in classes forms a tree.
7. There are 7 built-in metaclasses (of which only 5 are non-equivalent):
   • rdfs:Class, owl:Class, rdfs:Datatype, owl:DataRange, owl:Restriction, owl:DeprecatedClass, and owl:Nothing.

S1_o: The ontological structure of object membership

• Objects can have multiple minimum primary containers. That is, for an object x, existence of x.class is not guaranteed.
• In addition to terminal objects, there can be other objects (called properties) that are not descendants of helix objects.
• Inheritance does not have to be antisymmetric – classes and properties can have distinct equivalents.
• The only finiteness condition required is that helix eigenclasses, when ordered by reversed inheritance, are isomorphic to natural numbers.

An S1_o structure is a structure (O, ϵ, r, c, p) where

• O is a set of objects,
• ϵ is a relation between objects, called (object) membership,
• r is a distinuished object, the inheritance root,
• c is a distinuished object, the class classifier (instance root, metaclass root),
• p is a distinuished object, the property classifier.

• We write ϶ for the inverse of ϵ. We write x.϶ / x.ϵ for the set of members / containers of x (the preimage / image of {x} under ϵ).
• The inheritance, ≤ is a relation on O defined by: x ≤ y iff y.ϵ ⊆ x.ϵ.
• The usual inheritance ancestor/descendant terminology and notation (.↥/.↧) applies.
• The instance-of relation is the range-restriction of ϵ to classes.
• The eigenclass map, .ec, is defined in (S1_o~1).
• Eigenclasses are elements of O.ec. The remaining objects are primary.
• Helix objects, H, are objects x such that x ϵ x. By (S1_o~1), H.ec ⊆ H.
• Classes, C, are primary instances of c.
• Properties, P, are primary instances of p.
• Terminals, T, are primary objects that are neither classes nor properties, i.e. O = T ⊎ P ⊎ C ⊎ O.ec.
• Metaclasses are the lower bounds of helix classes, i.e. objects x such that H ∩ C ⊆ x.↥.

Note: Disjointness of P and C is asserted by (S1_o~10)(a).

Notes:

1. Requiring (S1_ϵ~5⁺) instead of (S1_ϵ~5) simplifies condition (S1_o~4)(d) – it follows that r.ec(i) is the only child of c.ec(i-1) for each i > 0.
2. As to the examples of structures disallowed by (S1_p~6), condition (S1_o~6) allows (ⅰ) but still disallows (ⅱ). []

Proposition:

1. For an S1_o structure (O, ϵ, r, c, p) the reduct (O, ϵ) is an S1_ϵ structure if and only if the following are satisfied:
   1. The set P of properties forms an antichain in inheritance.
   2. The eigenclass map, .ec, is injective.
   3. The class map, .class, is defined for every object.
   4. The set C of classes is finite.
2. For an S1_ϵ structure (O, ϵ) satisfying (S1_ϵ~5⁺) and containing at least one class p not from c.↥ ∪ c.↧, the structure (O, ϵ, r, c, p) is an S1_o structure.
3. Up to equivalence of inheritance roots and of metaclass roots, the structure is uniquely given by (O, ϵ, p).

Examples

ex:A rdf:type rdfs:Datatype .
ex:B rdf:type rdfs:Datatype .
ex:x rdf:type ex:A, ex:B .

ex:a rdf:type ex:Q .
ex:a rdfs:subPropertyOf ex:b .
ex:b rdfs:subPropertyOf ex:a .
ex:x rdfs:subPropertyOf ex:s .
ex:y rdfs:subPropertyOf ex:s .

S1_o.pr: The primary structure

• O.pr is set of primary objects.
• ϵ is a relation between primary objects, called (object) membership.
• ≤ is a relation between primary objects, called inheritance,
• r, c and p are distinguished primary objects.

• For the inheritance, the usual ancestor/descendant terminology and notation applies.
• For primary objects x, y, if x ϵ y then x is an instance-of y and y is a container-of x.
• Helix classes, H.pr, are primary objects x such that x ϵ x.
• Classes, C, are primary inheritance descendants of helix classes.
• Properties, P, are primary instances of p.
• Terminals, T, are primary objects that are neither classes nor properties, i.e. O.pr = T ⊎ P ⊎ C.
• Metaclasses are objects x such that H.pr ⊆ x.↥.
• We write ϵⁱ for i-th composition of ϵ, i > 0. In particular, (ϵ²) = (ϵ) ○ (ϵ). We put ϵ⁰ equal to the inheritance ≤.

Note: Disjointness of P and C is asserted by (S1_o.pr~9).

Eigenclass completion

Proposition: Up to isomorphism, there is one-to-one correspondence between S1_o structures and S1_o.pr structures.

Correspondence to RDFS₁ structures

• x ≤ y   iff   x = y or x ≤_C y or x ≤_P y.

Proposition A:

Proof: Acyclic chains in ϵ are of limited length due to the finiteness condition (RDFS₁~5).

Proposition B:

References

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