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 &lt;title&gt;S-TEN D2.1: Physical quantity, property and scale&lt;/title&gt;
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&lt;a name="physical_quantity_property_and_scale"/&gt;
&lt;h2&gt;7 Physical quantity, property and scale&lt;/h2&gt;

&lt;a name="physical_quantity_unit_and_scale"/&gt;
&lt;h3&gt;7.1 Physical quantity, unit and scale&lt;/h3&gt;

&lt;a name="number_with_unit"/&gt;
&lt;h3&gt;7.1.1 Number with unit&lt;/h3&gt;
&lt;p&gt;The path to a formal approach to the recording of physical quantities
measure begins with the concept of "number with unit". This is
a number with some information about what it means. For example we may have:&lt;/p&gt;

&lt;pre&gt;
[  a              myOnt:number_with_unit ;
   myOnt:number   "27.0"    ;
   myOnt:unit     bipm:metre  }
&lt;/pre&gt;

&lt;p class="note"&gt;NOTE Examples are given in
&lt;a href="http://www.w3.org/DesignIssues/Notation3"&gt;N3&lt;/a&gt;. They should be readable
even if you do not know the format.&lt;/p&gt;

&lt;p&gt;The problem with "number with unit" is that its meaning is vague. It is not
just a number, but is it a length
or an information object which describes a length?&lt;/p&gt;

&lt;h3&gt;7.1.2 Physical quantity and description&lt;/h3&gt;
&lt;p&gt;The physical quantity "27.0 metres" is a length.
Length is a "category of comparable physical quantities". (This phrase is taken
from the introduction to ISO 31-0.)&lt;/p&gt;

&lt;p&gt;The physical quantity "27.0 metres" may be described by an information object that consists
of a number and a unit. For example we may have:&lt;/p&gt;

&lt;pre&gt;
[  a          iso31:length ;
   myOnt:described_by
              [ a              myOnt:physical_quantity_description  ;
                myOnt:number   "27.0" ;
                myOnt:unit     bipm:metre ] ]
&lt;/pre&gt;

&lt;p&gt;There is nothing wrong with this approach, but it ignores the mathematics of
the relationship between
categories of physical quantities and numbers. As a result we have to invent new
information objects for all
related concepts such as length intervals, probability distributions over length,
etc..&lt;/p&gt;

&lt;h3&gt;7.1.3 Number&lt;/h3&gt;
&lt;p&gt;In the examples so far, we have regarded numbers as just literals. If we wish to use mathermatics it is
necessary to recognise that:&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;A number is an object. A number can have a representation in decimal or hex. Some numbers have names, such
as &pi;.&lt;/li&gt;
&lt;li&gt;A number space consists of a set of numbers, and relationships between the numbers which define the structure
of the space. A number space is more than just a set.&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;A number space, such as the reals, can be regarded as an &lt;b&gt;owl:Class&lt;/b&gt;. Hence
the literal "27" in the examples above can be
replaced by the anonymous object:&lt;/p&gt;

&lt;pre&gt;
[  a              sten:real ;
   basics:decimal   "27.0"  ]
&lt;/pre&gt;

&lt;p&gt;or more concisely, just:&lt;/p&gt;

&lt;pre&gt;
[  basics:decimal   "27.0"  ]
&lt;/pre&gt;

&lt;h3&gt;7.1.4 Scale&lt;/h3&gt;
&lt;p&gt;A scale is a function from a "category of comparable physical quantities"
(or &lt;a href="#physical_quantity_space"&gt;&lt;b&gt;sten:physical_quantity_space&lt;/b&gt;&lt;/a&gt;) to
a number space. A &lt;a href="#physical_quantity_space"&gt;&lt;b&gt;sten:physical_quantity_space&lt;/b&gt;&lt;/a&gt; consists of a set of
physical quantities, and relationships between the
physical quantities which define the structure of the space.&lt;/p&gt;

&lt;p&gt;A scale has properties (such as being increasing or continuous) which depend upon the structures of its domain
and range.&lt;/p&gt;

&lt;p&gt;An ontology can define &lt;b&gt;celsius&lt;/b&gt; and &lt;b&gt;metre_scale&lt;/b&gt;, as follows:&lt;/p&gt;

&lt;pre&gt;
bipm:celsius  a            owl:FunctionalProperty;
              rdfs:domain  iso31:thermodynamic_temperature  ;
              rdfs:range   sten:real  .

:metre_scale  a            owl:FunctionalProperty;
              rdfs:domain  iso31:length  ;
              rdfs:range   sten:real  .
&lt;/pre&gt;

&lt;p&gt;Using an ontology of scales, the representation of a physical quantity value is simple, rigorous and concise.
For the length 27.0 we have:&lt;/p&gt;

&lt;pre&gt;
[  a             iso31:length ;
   :metre_scale  [  basics:decimal  "27.0"  ]  ]
&lt;/pre&gt;

&lt;p&gt;or more concisely:&lt;/p&gt;

&lt;pre&gt;
[  :metre_scale  [  basics:decimal  "27.0"  ]  ]
&lt;/pre&gt;

&lt;p&gt;Similarly the representation of the temperature 20 degrees Celsius is:&lt;/p&gt;

&lt;pre&gt;
[  bipm:celsius  [  basics:decimal  "20.0"  ]  ]
&lt;/pre&gt;

&lt;h3&gt;7.1.5 Scale and unit&lt;/h3&gt;
&lt;p&gt;The suffix "_scale" has been included in the ID "metre_scale", because the SI concept of metre is not a scale
but a named length. In an ontology, the metre would be:&lt;/p&gt;

&lt;pre&gt;
bipm:metre    a             iso31:length  ;
              rdfs:comment  "Defined by the BIPM."  .
&lt;/pre&gt;

&lt;p&gt;Because metre_scale (like all other scales derived from units) is linear, the metre_scale is completely
defined by the statements:&lt;/p&gt;

&lt;pre&gt;
:metre_scale  a             sten:linear_function  .

bipm:metre   :metre_scale  [  basics:decimal  "1.0"  ]  .
&lt;/pre&gt;

&lt;p&gt;The standards on units are not good at distinguishing between units and scales. It is clear that all the
SI base units are indeed units and not scales.  The only two scales in common use are Celsius and
Fahrenheit.&lt;/p&gt;

&lt;p class="note"&gt;NOTE The Kelvin is a unit. It is a named thermodynamic temperature.
The Celsius is not a unit.&lt;/p&gt;

&lt;p&gt;The temperature scale &lt;a href="http://www.its-90.com"&gt;ITS-90&lt;/a&gt; is a scale. It
is a practical approximation to the scale that
is derived from the Kelvin unit.&lt;/p&gt;

&lt;a name="physical_quantity_and_property"/&gt;
&lt;h3&gt;7.2 Physical quantity and property&lt;/h3&gt;

&lt;a name="member_of_a_physical_quantity"/&gt;
&lt;h3&gt;7.2.1 Member of a physical quantity&lt;/h3&gt;

&lt;p&gt;A physical quantity can be regarded as a class.&lt;/p&gt;

&lt;div class="note"&gt;
&lt;p&gt;EXAMPLE 1 The physical quantity "27.0 metres" is a class which consists of
pairs of points in space.&lt;/p&gt;

&lt;p&gt;The ship "Jolly George" with a waterline length of 27 metres is not a member of
this class, but the pair of points defined by
the intersection of:&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;the surface of the sea at 2007-04-02T10:30;&lt;/li&gt;
&lt;li&gt;the plane of symmetry of the Jolly George at 2007-04-02T10:30;&lt;/li&gt;
&lt;li&gt;the outside surface of the the Jolly George at 2007-04-02T10:30,&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;is.&lt;/p&gt;
&lt;/div&gt;

&lt;p&gt;As in example 1, the members of a physical quantity are almost
invariably theoretical things rather than observable physical things.&lt;/p&gt;

&lt;div class="note"&gt;
&lt;p&gt;EXAMPLE 2 The physical quantity "10 tonnes" is a class which consists of
quantities of matter at an instant.&lt;/p&gt;

&lt;p&gt;Pump P_101 at 2007-04-02T10:30 is a member of the physical quantity
"10 tonnes", but this is not measurable. All we can say is that the average mass
during the measuring process, which lasts from 2007-04-02T10:29:59 to
2007-04-02T10:30:01 is "10 tonnes".&lt;/p&gt;
&lt;/div&gt;

&lt;div class="note"&gt;
&lt;p&gt;EXAMPLE 3 The physical quantity "7.1 tonnes per cubic metre" is a class which
consists of points within volumes of matter at an instant. This is a fictional
class, because matter is not a continuum and there is no concept of density at
sufficiently small scales.&lt;/p&gt;

&lt;p&gt;We cannot say that point P within the casing of pump P_101 at 2007-04-02T10:30
is a member of the class "7.1 tonnes per cubic metre". All we can say is that the
average density of a small volume surrounding point P for a small time interval
around 2007-04-02T10:30 is "7.1 tonnes per cubic metre".&lt;/p&gt;
&lt;/div&gt;

&lt;a name="relationship_with_a_physical_quantity"/&gt;
&lt;h3&gt;7.2.2 Relationship with a physical quantity&lt;/h3&gt;

&lt;p&gt;A physical aspect of a &lt;a href="physical_things.htm#physical_thing"&gt;&lt;b&gt;sten:physical_thing&lt;/b&gt;&lt;/a&gt; can be recorded as a
relationship with a &lt;a href="#physical_quantity"&gt;&lt;b&gt;sten:physical_quantity&lt;/b&gt;&lt;/a&gt;. The &lt;a href="physical_things.htm#physical_thing"&gt;&lt;b&gt;sten:physical_thing&lt;/b&gt;&lt;/a&gt;
is not necessarily a member of the &lt;a href="#physical_quantity"&gt;&lt;b&gt;sten:physical_quantity&lt;/b&gt;&lt;/a&gt;.&lt;/p&gt;

&lt;div class="note"&gt;
&lt;p&gt;EXAMPLE 1 The statement "The Jolly George at 2007-04-02T10:30
has a waterline length of exactly 27 metres", which is a valid statement but which
is probably untrue, can be recorded as follows:&lt;/p&gt;

&lt;pre&gt;
:Jolly_George_at_2007-04-02T10:30
        a                      sten:classical_object_at_instant ;
        sten:temporal_part_of  :Jolly_George ;
        ship:waterline_length  [ :metre_scale [basics:decimal "27.0" ] .
&lt;/pre&gt;

&lt;p&gt;The pair of points which defines the waterline length of the Jolly George
(see the example in clause
&lt;a href="#member_of_a_physical_quantity"&gt;Member of a physical quantity&lt;/a&gt;)
is a member of "27 metres", but not the ship.&lt;/p&gt;
&lt;/div&gt;

&lt;p&gt;The relationship is an evaluation of a &lt;a href="#physical_property"&gt;&lt;b&gt;sten:physical_property&lt;/b&gt;&lt;/a&gt;.
This is a &lt;b&gt;basics:function&lt;/b&gt;
which has a space of instances of &lt;a href="physical_things.htm#physical_thing"&gt;&lt;b&gt;sten:physical_thing&lt;/b&gt;&lt;/a&gt; as its domain and a
&lt;a href="#physical_quantity_space"&gt;&lt;b&gt;sten:physical_quantity_space&lt;/b&gt;&lt;/a&gt; as its range.&lt;/p&gt;

&lt;div class="note"&gt;
&lt;p&gt;EXAMPLE 2 The &lt;b&gt;basics:function&lt;/b&gt; &lt;b&gt;ship:waterline_length&lt;/b&gt;
has instants in the life of ships as its domain and &lt;b&gt;iso31:length&lt;/b&gt; as
its range. This can be recorded as follows:&lt;/p&gt;

&lt;pre&gt;
ship:waterline_length a             owl:FunctionalProperty ;
                      rdfs:domain
                          [ owl:intersectionOf
                              ( sten:classical_object_at_instant
                                [ owl:onProperty     sten:temporal_part_of ;
                                  owl:allValuesFrom  ship:ship ]
                              )
                          ] ;
                      rdfs:range  iso31:length .
&lt;/pre&gt;
&lt;/div&gt;

&lt;p&gt;If the evaluation of &lt;a href="#physical_property"&gt;&lt;b&gt;sten:physical_property&lt;/b&gt;&lt;/a&gt; is uncertain, then a
region which contains the value can be specified.&lt;/p&gt;

&lt;div class="note"&gt;
&lt;p&gt;EXAMPLE 3 A more truthful version of the statement in EXAMPLE 1 is
"The Jolly George at 2007-04-02T10:30
has a waterline length which is within an understood tolerance of 27 metres", where
"understood tolerance" will be interpreted according to a activity.
This can be recorded as follows:&lt;/p&gt;

&lt;pre&gt;
:Jolly_George_at_2007-04-02T10:30
        a                      sten:classical_object_at_instant ;
        sten:temporal_part_of  :Jolly_George ;
        ship:waterline_length  [  rdf:type
                                  [ rdfs:subClassOf  iso31:length ;
                                    rdf:type         sten:understood_tolerance ;
                                    sten:has_nominal_value
                                       [ :metre_scale [basics:decimal "27.0" ]
                                  ]
                               ] .
&lt;/pre&gt;
&lt;/div&gt;

&lt;p&gt;In order to record the variation of a &lt;a href="#physical_property"&gt;&lt;b&gt;sten:physical_property&lt;/b&gt;&lt;/a&gt; with time,
a sub-property can be defined that has a 1D manifold of temporal instants as
its domain. The use of functions to record property variation is discussed in
&lt;a href="temporal_part_and_time.htm#an_approach_to_property_variation"&gt;An approach to property variation&lt;/a&gt;.&lt;/p&gt;

&lt;div class="note"&gt;
&lt;p&gt;EXAMPLE 4 The variation of the waterline length of the Jolly George can
be recorded by the "Jolly George waterline length" function. The domain of this
function is the life of the Jolly George regarded as a 1D manifold of instants,
and the range is &lt;b&gt;iso31:length&lt;/b&gt;. The image
of this function is within an understood tolerance of 27 metres. This
can be recorded as follows:&lt;/p&gt;

&lt;pre&gt;
:Jolly_George    a                ship:ship ;
                 a                basics:manifold_1d ;
                 rdfs:subClassOf  sten:classical_object_at_instant .

[ rdfs:domain         :Jolly_George         ;
  rdfs:range          iso31:length          ;
  rdfs:subPropertyOf  ship:waterline_length ]
                sten:has_image    [ rdfs:subClassOf  iso31:length ;
                                    rdf:type         sten:understood_tolerance ;
                                    sten:has_nominal_value
                                       [ :metre_scale [basics:decimal "27.0" ]
                                  ] .
&lt;/pre&gt;
&lt;/div&gt;

&lt;p&gt;In the case of a fundamental property such as "mass", there is no
&lt;a href="#physical_property"&gt;&lt;b&gt;sten:physical_property&lt;/b&gt;&lt;/a&gt;. Instead, the relationship between
the &lt;a href="physical_things.htm#physical_thing"&gt;&lt;b&gt;sten:physical_thing&lt;/b&gt;&lt;/a&gt; and the &lt;a href="#physical_quantity"&gt;&lt;b&gt;sten:physical_quantity&lt;/b&gt;&lt;/a&gt;
is classification. This causes a practical
problem because classification is not a function. You cannot ask "what is a
thing classified as?" and expect a sensible answer.&lt;/p&gt;

&lt;p&gt;This problem can be avoided by deriving a trivial
&lt;a href="#physical_property"&gt;&lt;b&gt;sten:physical_property&lt;/b&gt;&lt;/a&gt; from each
&lt;a href="#physical_quantity_space"&gt;&lt;b&gt;sten:physical_quantity_space&lt;/b&gt;&lt;/a&gt;.&lt;/p&gt;

&lt;div class="note"&gt;
&lt;p&gt;NOTE This seems very sordid, and it would be good to have an alternative.&lt;/p&gt;
&lt;/div&gt;

&lt;div class="note"&gt;
&lt;p&gt;EXAMPLE 5 The variation of the mass of the Jolly George can
be recorded by the "Jolly George mass" function, in a way that is analogous to
EXAMPLE 4. The domain of this
function is the life of the Jolly George regarded as a 1D manifold of instants,
and the range is &lt;b&gt;iso31:mass&lt;/b&gt;. The image
of this function is within an understood tolerance of 63 tonnes. This
can be recorded as follows:&lt;/p&gt;

&lt;pre&gt;
:Jolly_George    a                ship:ship ;
                 a                basics:manifold_1d ;
                 rdfs:subClassOf  sten:classical_object_at_instant .

[ rdfs:domain         :Jolly_George         ;
  rdfs:range          iso31:mass          ;
  rdfs:subPropertyOf  sten:has_mass ]
                sten:has_image    [ rdfs:subClassOf  iso31:mass ;
                                    rdf:type         sten:understood_tolerance ;
                                    sten:has_nominal_value
                                       [ :tonne_scale [basics:decimal "63.0" ]
                                  ] .
&lt;/pre&gt;

&lt;p&gt;The function &lt;b&gt;has:mass&lt;/b&gt; is defined as follows:&lt;/p&gt;

&lt;pre&gt;
has_mass    a                   owl:FunctionalProperty ;
            rdfs:domain         sten:physical_thing    ;
            rdfs:range          iso31:mass             ;
            rdfs:subPropertyOf  rdf:type               .
&lt;/pre&gt;
&lt;/div&gt;

&lt;a name="ontology_for_physical_quantity_unit_and_scale"/&gt;
&lt;h3&gt;7.2 Ontology for physical quantity, unit and scale&lt;/h3&gt;

&lt;a name="physical_quantity"/&gt;
&lt;h3&gt;7.2.1 physical_quantity&lt;/h3&gt;
&lt;p&gt;An object is a &lt;a href="#physical_quantity"&gt;&lt;b&gt;sten:physical_quantity&lt;/b&gt;&lt;/a&gt; if and only if
it is a "physical quantity", where the term is as used in ISO 31-0.&lt;/p&gt;

&lt;div class="note"&gt;
&lt;p&gt;NOTE 1 ISO 31-0 does not define the term "physical quantity". A possible definition is
"magnitude of a physical phenomenon, which can be compared with other
magnitudes of the same physical phenomenon."&lt;/p&gt;
&lt;/div&gt;

&lt;div class="note"&gt;
&lt;p&gt;NOTE 2 Units are named instances of &lt;a href="#physical_quantity"&gt;&lt;b&gt;sten:physical_quantity&lt;/b&gt;&lt;/a&gt;.&lt;/p&gt;
&lt;/div&gt;

&lt;div class="note"&gt;
&lt;p&gt;EXAMPLE&lt;/p&gt;
&lt;pre&gt;
bipm:metre   a  sten:physical_quantity ;
             a  iso31:length .
&lt;/pre&gt;
&lt;/div&gt;

&lt;p&gt;&lt;u&gt;OWL specification:&lt;/u&gt;&lt;/p&gt;
&lt;pre&gt;
-->
<owl:Class rdf:about="&sten;physical_quantity">
 <rdfs:subClassOf rdf:resource="&basics;structure_point"/>
 <owl:sameAs rdf:resource="&iso15926-2;property"/>
 <meta:defined_by rdf:resource="&D2.1;/physical_quantity_property_and_scale.htm#physical_quantity"/>
</owl:Class>
<!--
&lt;/pre&gt;

&lt;a name="physical_quantity_space"/&gt;
&lt;h3&gt;7.2.2 physical_quantity_space&lt;/h3&gt;
&lt;p&gt;An object is a &lt;a href="#physical_quantity_space"&gt;&lt;b&gt;sten:physical_quantity_space&lt;/b&gt;&lt;/a&gt; if and only if
it is a "category of physical quantities which are mutually comparable".&lt;/p&gt;

&lt;div class="note"&gt;
&lt;p&gt;NOTE 1 The clause "category of physical quantities which are mutually comparable"
is as used in ISO 31-0, but the object is not assigned a name.&lt;/p&gt;
&lt;/div&gt;

&lt;div class="note"&gt;
&lt;p&gt;NOTE 2 Instances of &lt;a href="#physical_quantity_space"&gt;&lt;b&gt;sten:physical_quantity_space&lt;/b&gt;&lt;/a&gt; are defined in the parts
of ISO 31. These instances have been recorded as OWL within STEN deliverable D3.2.&lt;/p&gt;
&lt;/div&gt;

&lt;div class="note"&gt;
&lt;p&gt;EXAMPLE 1&lt;/p&gt;
&lt;pre&gt;
iso31:thermodynamic_temperature  a  sten:physical_quantity_space .
&lt;/pre&gt;
&lt;/div&gt;

&lt;div class="note"&gt;
&lt;p&gt;EXAMPLE 2&lt;/p&gt;
&lt;pre&gt;
iso31:mass  a  sten:physical_quantity_space .
&lt;/pre&gt;
&lt;/div&gt;

&lt;p&gt;&lt;u&gt;OWL specification:&lt;/u&gt;&lt;/p&gt;
&lt;pre&gt;
-->
<owl:Class rdf:about="&sten;physical_quantity_space">
 <rdfs:subClassOf rdf:resource="&basics;structure"/>
 <owl:sameAs rdf:resource="&iso15926-2;property_space"/>
 <meta:defined_by rdf:resource="&D2.1;/physical_quantity_property_and_scale.htm#physical_quantity_space"/>
</owl:Class>
<!--
&lt;/pre&gt;

&lt;a name="scale"/&gt;
&lt;h3&gt;7.2.3 scale&lt;/h3&gt;
&lt;p&gt;An object is a &lt;a href="#scale"&gt;&lt;b&gt;sten:scale&lt;/b&gt;&lt;/a&gt; if and only if
it is a &lt;b&gt;basics:function&lt;/b&gt; from a &lt;a href="#physical_quantity_space"&gt;&lt;b&gt;sten:physical_quantity_space&lt;/b&gt;&lt;/a&gt;
to a numeric space.&lt;/p&gt;

&lt;div class="note"&gt;
&lt;p&gt;EXAMPLE 1&lt;/p&gt;
&lt;pre&gt;
bipm:celsius  a           sten:scale ;
              rdfs:domain iso31:thermodynamic_temperature .
              rdfs:range  basics:real_numbers .
&lt;/pre&gt;
&lt;/div&gt;

&lt;p&gt;&lt;u&gt;OWL specification:&lt;/u&gt;&lt;/p&gt;
&lt;pre&gt;
-->
<owl:Class rdf:about="&sten;scale">
 <rdfs:subClassOf rdf:resource="&owl;FunctionalProperty"/>
 <rdfs:subClassOf>
  <owl:Restriction>
   <owl:onProperty rdf:resource="&rdfs;domain"/>
   <owl:allValuesFrom rdf:resource="&sten;physical_quantity_space"/>
  </owl:Restriction>
 </rdfs:subClassOf>
 <owl:sameAs rdf:resource="&iso15926-2;scale"/>
 <meta:defined_by rdf:resource="&D2.1;/physical_quantity_property_and_scale.htm#scale"/>
</owl:Class>
<!--
&lt;/pre&gt;

&lt;div class="note"&gt;
&lt;p&gt;NOTE The restriction on the range is a bit tricky. Probably is any
structure. For the scale to be useful, there has to be a way of identifying each member
of the structure.&lt;/p&gt;
&lt;/div&gt;

&lt;a name="linear_function"/&gt;
&lt;h3&gt;7.2.4 linear_function&lt;/h3&gt;
&lt;p&gt;An object is a &lt;a href="#linear_function"&gt;&lt;b&gt;sten:linear_function&lt;/b&gt;&lt;/a&gt; if and only if:&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;it is a &lt;b&gt;basics:function&lt;/b&gt; (&lt;b&gt;F&lt;/b&gt;) between two instances of
&lt;b&gt;basics:vector_space&lt;/b&gt; over the same scalars, or between a
&lt;b&gt;basics:vector_space&lt;/b&gt; and itself;&lt;/li&gt;
&lt;li&gt;the function is such that:&lt;/li&gt;
&lt;/ul&gt;

&lt;p class="caption"&gt;&lt;b&gt;F&lt;/b&gt;(k.&lt;b&gt;&xlt;/b&gt;) = k.&lt;b&gt;F&lt;/b&gt;(&lt;b&gt;&xlt;/b&gt;)&lt;/p&gt;

&lt;p&gt;where &lt;b&gt;&xlt;/b&gt; is in the domain of &lt;b&gt;F&lt;/b&gt;, and k is a scalar.&lt;/p&gt;

&lt;p&gt;&lt;u&gt;OWL specification:&lt;/u&gt;&lt;/p&gt;
&lt;pre&gt;
-->
<owl:Class rdf:about="&sten;linear_function">
 <rdfs:subClassOf rdf:resource="&owl;FunctionalProperty"/>
 <rdfs:subClassOf>
  <owl:Restriction>
   <owl:onProperty rdf:resource="&rdfs;domain"/>
   <owl:allValuesFrom rdf:resource="&basics;vector_space"/>
  </owl:Restriction>
 </rdfs:subClassOf>
 <rdfs:subClassOf>
  <owl:Restriction>
   <owl:onProperty rdf:resource="&rdfs;range"/>
   <owl:allValuesFrom rdf:resource="&basics;vector_space"/>
  </owl:Restriction>
 </rdfs:subClassOf>
 <meta:defined_by rdf:resource="&D2.1;/physical_quantity_property_and_scale.htm#linear_function"/>
</owl:Class>
<!--
&lt;/pre&gt;


&lt;a name="physical_property"/&gt;
&lt;h3&gt;7.2.5 physical_property&lt;/h3&gt;
&lt;p&gt;An object is a &lt;a href="#physical_property"&gt;&lt;b&gt;sten:physical_property&lt;/b&gt;&lt;/a&gt; if and only if&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;it is a &lt;b&gt;basics:function&lt;/b&gt; from a subclass of
&lt;a href="physical_things.htm#physical_thing"&gt;&lt;b&gt;sten:physical_thing&lt;/b&gt;&lt;/a&gt; to a &lt;a href="#physical_quantity_space"&gt;&lt;b&gt;sten:physical_quantity_space&lt;/b&gt;&lt;/a&gt;;&lt;/li&gt;
&lt;li&gt;the &lt;a href="#physical_quantity"&gt;&lt;b&gt;sten:physical_quantity&lt;/b&gt;&lt;/a&gt; is a physical aspect of the
&lt;a href="physical_things.htm#physical_thing"&gt;&lt;b&gt;sten:physical_thing&lt;/b&gt;&lt;/a&gt;.&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;&lt;u&gt;OWL specification:&lt;/u&gt;&lt;/p&gt;
&lt;pre&gt;
-->
<owl:Class rdf:about="&sten;physical_property">
 <rdfs:subClassOf rdf:resource="&owl;FunctionalProperty"/>
 <rdfs:subClassOf>
  <owl:Restriction>
   <owl:onProperty rdf:resource="&rdfs;range"/>
   <owl:allValuesFrom rdf:resource="&sten;physical_quantity_space"/>
  </owl:Restriction>
 </rdfs:subClassOf>
 <owl:sameAs rdf:resource="&iso15926-2;indirect_property"/>
 <meta:defined_by rdf:resource="&D2.1;/physical_quantity_property_and_scale.htm#physical_property"/>
</owl:Class>
<!--
&lt;/pre&gt;

&lt;div class="note"&gt;
&lt;p&gt;NOTE The restriction on the domain is a bit tricky. It is any
subclass of &lt;a href="physical_things.htm#physical_thing"&gt;&lt;b&gt;sten:physical_thing&lt;/b&gt;&lt;/a&gt;.&lt;/p&gt;
&lt;/div&gt;

&lt;a name="understood_tolerance"/&gt;
&lt;h3&gt;7.2.6 understood_tolerance&lt;/h3&gt;
&lt;p&gt;An object is a &lt;a href="#understood_tolerance"&gt;&lt;b&gt;sten:understood_tolerance&lt;/b&gt;&lt;/a&gt; if and only if&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;it is &lt;b&gt;basics:substructure&lt;/b&gt;;&lt;/li&gt;
&lt;li&gt;it is a tolerance about a nominal value which is interpreted according to
an activity.&lt;/li&gt;
&lt;/ul&gt;


&lt;div class="note"&gt;
&lt;p&gt;NOTE An &lt;a href="#understood_tolerance"&gt;&lt;b&gt;sten:understood_tolerance&lt;/b&gt;&lt;/a&gt;
cam be within a
&lt;a href="#physical_quantity_space"&gt;&lt;b&gt;sten:physical_quantity_space&lt;/b&gt;&lt;/a&gt;.&lt;/p&gt;

&lt;p&gt;An &lt;a href="#understood_tolerance"&gt;&lt;b&gt;sten:understood_tolerance&lt;/b&gt;&lt;/a&gt;
can be a &lt;a href="temporal_part_and_time.htm#period"&gt;&lt;b&gt;sten:period&lt;/b&gt;&lt;/a&gt;.&lt;/p&gt;

&lt;p&gt;An &lt;a href="#understood_tolerance"&gt;&lt;b&gt;sten:understood_tolerance&lt;/b&gt;&lt;/a&gt;
can be a &lt;a href="temporal_part_and_time.htm#region"&gt;&lt;b&gt;sten:region&lt;/b&gt;&lt;/a&gt;.&lt;/p&gt;
&lt;/div&gt;

&lt;p&gt;&lt;u&gt;OWL specification:&lt;/u&gt;&lt;/p&gt;
&lt;pre&gt;
-->
<owl:Class rdf:about="&sten;understood_tolerance">
 <rdfs:subClassOf rdf:resource="&basics;sub_structure"/>
 <meta:defined_by rdf:resource="&D2.1;/physical_quantity_property_and_scale.htm#understood_tolerance"/>
</owl:Class>
<!--
&lt;/pre&gt;

&lt;a name="has_nominal_value"/&gt;
&lt;h3&gt;7.2.7 has_nominal_value&lt;/h3&gt;
&lt;p&gt;An object is a &lt;a href="#has_nominal_value"&gt;&lt;b&gt;sten:has_nominal_value&lt;/b&gt;&lt;/a&gt; relationship if and only if:&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;it is a &lt;b&gt;basics:function&lt;/b&gt; from
&lt;a href="#understood_tolerance"&gt;&lt;b&gt;sten:understood_tolerance&lt;/b&gt;&lt;/a&gt;
to  &lt;b&gt;basics:structure_point&lt;/b&gt;;&lt;/li&gt;
&lt;li&gt;it is a nominal value around which a tolerance is understood.&lt;/li&gt;
&lt;/ul&gt;

&lt;div class="note"&gt;
&lt;p&gt;NOTE If an &lt;a href="#understood_tolerance"&gt;&lt;b&gt;sten:understood_tolerance&lt;/b&gt;&lt;/a&gt;
is within a
&lt;a href="#physical_quantity_space"&gt;&lt;b&gt;sten:physical_quantity_space&lt;/b&gt;&lt;/a&gt;
then a &lt;a href="#has_nominal_value"&gt;&lt;b&gt;sten:has_nominal_value&lt;/b&gt;&lt;/a&gt;
gives a &lt;a href="#physical_quantity"&gt;&lt;b&gt;sten:physical_quantity&lt;/b&gt;&lt;/a&gt;.&lt;/p&gt;

&lt;p&gt;If an &lt;a href="#understood_tolerance"&gt;&lt;b&gt;sten:understood_tolerance&lt;/b&gt;&lt;/a&gt;
is within a
&lt;a href="temporal_part_and_time.htm#time"&gt;&lt;b&gt;sten:time&lt;/b&gt;&lt;/a&gt;
then a &lt;a href="#has_nominal_value"&gt;&lt;b&gt;sten:has_nominal_value&lt;/b&gt;&lt;/a&gt;
gives an &lt;a href="temporal_part_and_time.htm#instant"&gt;&lt;b&gt;sten:instant&lt;/b&gt;&lt;/a&gt;.&lt;/p&gt;

&lt;p&gt;If an &lt;a href="#understood_tolerance"&gt;&lt;b&gt;sten:understood_tolerance&lt;/b&gt;&lt;/a&gt;
is within a
&lt;a href="temporal_part_and_time.htm#physical_space"&gt;&lt;b&gt;sten:physical_space&lt;/b&gt;&lt;/a&gt;
then a &lt;a href="#has_nominal_value"&gt;&lt;b&gt;sten:has_nominal_value&lt;/b&gt;&lt;/a&gt;
gives a &lt;a href="temporal_part_and_time.htm#point_in_space"&gt;&lt;b&gt;sten:point_in_space&lt;/b&gt;&lt;/a&gt;.&lt;/p&gt;
&lt;/div&gt;

&lt;p&gt;&lt;u&gt;OWL specification:&lt;/u&gt;&lt;/p&gt;
&lt;pre&gt;
-->
<owl:FunctionalProperty rdf:about="&sten;has_nominal_value">
 <rdfs:domain rdf:resource="&sten;understood_tolerance"/>
 <rdfs:range rdf:resource="&basics;structure_point"/>
 <meta:defined_by rdf:resource="&D2.1;/physical_quantity_property_and_scale.htm#has_nominal_value"/>
</owl:FunctionalProperty>
<!--
&lt;/pre&gt;

&lt;a name="has_image"/&gt;
&lt;h3&gt;7.2.8 has_image&lt;/h3&gt;
&lt;p&gt;An object is a &lt;a href="#has_image"&gt;&lt;b&gt;sten:has_image&lt;/b&gt;&lt;/a&gt; relationship if and only if:&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;it is a &lt;b&gt;basics:function&lt;/b&gt; from &lt;b&gt;basics:mapping&lt;/b&gt;
to &lt;b&gt;owl:Class&lt;/b&gt;;&lt;/li&gt;
&lt;li&gt;it is the &lt;b&gt;owl:Class&lt;/b&gt; that consists of all the values of the &lt;b&gt;basics:mapping&lt;/b&gt;
and nothing else.&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;&lt;u&gt;OWL specification:&lt;/u&gt;&lt;/p&gt;
&lt;pre&gt;
-->
<owl:FunctionalProperty rdf:about="&sten;has_image">
 <rdfs:domain rdf:resource="&basics;mapping"/>
 <rdfs:range rdf:resource="&owl;Class"/>
 <meta:defined_by rdf:resource="&D2.1;/physical_quantity_property_and_scale.htm#has_image"/>
</owl:FunctionalProperty>
<!--
&lt;/pre&gt;

&lt;/body&gt;
&lt;/html&gt;
-->

</rdf:RDF>