## Mathematics

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Last Update: 4/5/02
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### Main

#$MathematicalOrComputationalThing mathematical or computational objects

> The collection of #$Intangible things that are intrinsically mathematical (see #$MathematicalThing) or computational (see #$ComputationalObject). Instances of #$MathematicalOrComputationalThing are abstract in the very strong sense of being nonspatial, atemporal, and massless. Examples include numbers, sets, collections, relations, algorithms, and abstract character strings.
> guid: bd58e31f-9c29-11b1-9dad-c379636f7270
> direct instance of: #$VariableOrderCollection
> direct specialization of: #$Intangible
> direct generalization of: #$MathematicalThing #$Algorithm #$ComputationalObject

#$ComputationalObject computational objects **COMMENT NOT REVIEWED** **GAFs NOT REVIEWED**

> A specialization of both #$MathematicalOrComputationalThing and #$IntangibleIndividual. Each instance of #$ComputationalObject is a syntactically structured form, such as a Cyc system expression, a Lisp string, a C variable name, or an equation in a particular canonical form format.
> guid: bd58803b-9c29-11b1-9dad-c379636f7270
> direct instance of: #$ObjectType
> direct specialization of: #$MathematicalOrComputationalThing #$IntangibleIndividual
> direct generalization of: #$JustificationTruth #$RuleTemplate #$CycLAssertionDirection #$CycHLTruthValue #$ELSentenceTemplate

### Numbers

#$IntervalOnNumberLine intervals (quantities) **COMMENT NOT REVIEWED** **GAFs NOT REVIEWED**

> A specialization of #$ScalarInterval. Each instance of #$IntervalOnNumberLine is an interval on the real number line; for example, the interval described by `numbers greater than zero and less than or equal to 10'. A common special case of such intervals is that of a single point on that line, viz., a number such as five or 125. Note that such an interval need not be contiguous; e.g., `even numbers between Pi and the square root of 1000' describes a legitimate instance of #$IntervalOnNumberLine. The collection #$RealNumber is itself a specialization of #$IntervalOnNumberLine.
> guid: bd58f32f-9c29-11b1-9dad-c379636f7270
> direct instance of: #$MeasurableScalarIntervalType
> direct specialization of: #$ScalarInterval
> direct generalization of: #$IntegerExtent #$RealNumber

#$RealNumber real numbers **COMMENT NOT REVIEWED** **GAFs NOT REVIEWED**

> The collection of all the minimal intervals (i.e., points) on the number line; a subcollection of #$IntervalOnNumberLine. Each instance of #$RealNumber is a single point on the real number line, which has no upper or lower bounds. Subcollections of #$RealNumber include #$Integer, #$RationalNumber, #$NegativeNumber, #$PrimeNumber, and others. Note: Real numbers, like other instances of #$IntervalOnNumberLine, are measured along a single number `line'; but complex numbers, quaternions, etc., are n-tuples of numbers, and therefore are instances of #$NTupleInterval. For example, #$ComplexNumber is a subcollection of #$NTupleInterval
> guid: bd588098-9c29-11b1-9dad-c379636f7270
> direct instance of: #$MeasurableScalarIntervalType
> direct specialization of: #$ScalarPointValue #$IntervalOnNumberLine #$ComplexNumber
> direct generalization of: #$NonPositiveNumber #$NonNegativeNumber #$RationalNumber #$PositiveNumber #$NegativeNumber

#$RationalNumber rational numbers

> A specialization of #$RealNumber. A number NUM is an instance of #$RationalNumber just in case NUM can be expressed as the quotient of two integers. For example, 3/4, 2 1/8, 0.3333333..., 11/5.
> guid: bd58f2aa-9c29-11b1-9dad-c379636f7270
> direct instance of: #$MeasurableScalarIntervalType
> direct specialization of: #$RealNumber
> direct generalization of: #$SubLRealNumber #$Integer

#$Integer integers **COMMENT NOT REVIEWED** **GAFs NOT REVIEWED**

> The collection of all whole numbers; a subcollection of #$RationalNumber. Each instance of #$Integer is a whole number, resolvable into units with no fractional remainder. An integer may be positive (e.g., 42), zero, or negative (e.g., -42). Note that 42.0 is a floating-point real number which is close to the integer 42 within the tolerance of the floating-point representation but is not necessarily equal to the integer 42. Therefore, 42.0 is not an instance of #$Integer.
> guid: bd589e12-9c29-11b1-9dad-c379636f7270
> direct instance of: #$MeasurableScalarIntervalType
> direct specialization of: #$IntegerExtent #$RationalNumber
> direct generalization of: #$NonPositiveInteger #$OddNumber #$EvenNumber #$NegativeInteger #$SubLInteger #$PositiveInteger #$NonNegativeInteger

#$EvenNumber even numbers **COMMENT NOT REVIEWED** **GAFs NOT REVIEWED**

> The set of all even numbers (integers) including positive and negative even numbers and zero, but not including any infinite 'numbers'.
> guid: bff46760-9c29-11b1-9dad-c379636f7270
> direct instance of: #$MeasurableScalarIntervalType
> direct specialization of: #$Integer

#$OddNumber odd numbers **COMMENT NOT REVIEWED** **GAFs NOT REVIEWED**

> The set of all odd numbers (integers) including positive and negative odd numbers, but not including any infinite 'numbers'.
> guid: becb081f-9c29-11b1-9dad-c379636f7270
> direct instance of: #$MeasurableScalarIntervalType
> direct specialization of: #$Integer

#$PositiveNumber positive (number)

> A specialization of #$RealNumber. An instance NUMBER of #$RealNumber is also an instance of #$PositiveNumber just in case NUMBER is greater than zero. Instances of #$PositiveNumber include 42 and 0.17, but not 0 or -5.
> guid: bf6afb32-9c29-11b1-9dad-c379636f7270
> direct instance of: #$MeasurableScalarIntervalType
> direct specialization of: #$PositiveScalarInterval #$NonNegativeNumber
> direct generalization of: #$Real1-Infinity #$PositiveInteger

#$NegativeNumber negative numbers **GAFs NOT REVIEWED**

> A specialization of #$RealNumber. An instance REAL of #$RealNumber is also an instance of #$NegativeNumber just in case REAL is less than 0.
> guid: c0fe03e3-9c29-11b1-9dad-c379636f7270
> direct instance of: #$MeasurableScalarIntervalType
> direct specialization of: #$NegativeScalarInterval #$NonPositiveNumber
> direct generalization of: #$NegativeInteger

#$PositiveInteger positive integers

> A specialization of #$Integer. An instance INT of #$Integer is an instance of #$PositiveInteger just in case INT is an integer greater than 0.
> guid: bd589ed9-9c29-11b1-9dad-c379636f7270
> direct instance of: #$MeasurableScalarIntervalType
> direct specialization of: #$PositiveIntegerExtent #$PositiveNumber #$NonNegativeInteger

#$NegativeInteger negative integers **GAFs NOT REVIEWED**

> A specialization of #$Integer. An instance INT of #$Integer is also an instance of #$NegativeInteger if and only if INT is less than zero.
> guid: c0fe031d-9c29-11b1-9dad-c379636f7270
> direct instance of: #$MeasurableScalarIntervalType
> direct specialization of: #$NegativeIntegerExtent #$NonPositiveInteger #$NegativeNumber

#$NonNegativeNumber non-negative numbers

> #$NonNegativeNumber is the sub-collection of #$RealNumber that excludes the negative reals. Each instance of #$NonNegativeNumber is a number greater than or equal to zero -- for example, 0, 0.173, Pi, 4, and 101. Quantities measured in units -- for example, (#$SecondsDuration 4) and (#$Mile 42) -- are not instances of #$NonNegativeNumber, but rather are instances of its super-collection #$NonNegativeScalarInterval.
> guid: bd58f22b-9c29-11b1-9dad-c379636f7270
> direct instance of: #$MeasurableScalarIntervalType
> direct specialization of: #$NonNegativeScalarInterval #$RealNumber
> direct generalization of: #$PositiveNumber #$NonNegativeInteger

#$NonNegativeInteger non-negative integers

> #$NonNegativeInteger is the sub-collection of #$Integer that excludes the negative integers. Each instance of #$NonNegativeInteger is a whole number greater than or equal to zero -- for example, 0, 1, 2, 3, ....
> guid: bd58bf69-9c29-11b1-9dad-c379636f7270
> direct instance of: #$MeasurableScalarIntervalType
> direct specialization of: #$NonNegativeIntegerExtent #$Cardinal-Mathematical #$NonNegativeNumber #$Integer
> direct generalization of: #$CycUniversalSecond #$PositiveInteger

### Predicates

#$decodingDeviceType decoding device type **COMMENT NOT REVIEWED** **GAFs NOT REVIEWED**

> This predicate is used to specify the type of device needed for decoding information contained in a particular kind of encoding scheme. (#$decodingDeviceType SCHEME DEV-TYP) means that DEV-TYP is the type of device required to decode the information encoded with encoding scheme SCHEME. For example, a telephone is the kind of device needed to decode audio information sent over telephone lines. A cable TV box is the device type needed to decode cable TV signals. And so on. See also #$EncodingSchemeType.
> guid: bd58cef2-9c29-11b1-9dad-c379636f7270
> direct instance of: #$BinaryPredicate

#$deviceControlledBy device controlled by **COMMENT NOT REVIEWED** **GAFs NOT REVIEWED**

> (#$deviceControlledBy DEV CONTROL) means that CONTROL is a #$ControlDevice that controls #$PhysicalDevice DEV. Most of the time, the controls will be #$physicalParts of the device. However there are some notable exceptions -- one's TV remote control, for example -- which control a device but are not #$physicalParts of that device.
> guid: bd5cbb65-9c29-11b1-9dad-c379636f7270
> direct instance of: #$AsymmetricBinaryPredicate

#$instrument-Generic instrument (actor slot)

> The predicate #$instrument-Generic is used to link a particular event to any of the objects which play an instrumental role in it. (#$instrument-Generic EVENT OBJECT) means that OBJECT plays an intermediate causal role in EVENT, facilitating the occurrence of EVENT, and serving some purpose of some #$Agent. This can happen in at least two ways: either the `doer' of EVENT acts on OBJECT, which in turn acts on something else (as when someone uses a hammer to pound in a nail) or the `doer' of EVENT acts on something, making it possible for OBJECT to act on that thing (as when someone puts wet clothes out in the sun to dry). Typically, an #$instrument-Generic is not significantly altered by playing that role in an event. #$deviceUsed is an important specialization of #$instrument-Generic.
> guid: bd5d4eb3-9c29-11b1-9dad-c379636f7270
> direct instance of: #$IndividualLevelPredicate #$ActorSlot
> direct specialization of: #$actors #$instrumentalRole

#$deviceUsed device used (actor slot)

> The predicate #$deviceUsed relates an event to a device used in that event. (#$deviceUsed EVENT OBJECT) means that the #$PhysicalDevice OBJECT plays an instrumental role in the #$Event EVENT (see the more generalized predicate #$instrument-Generic), OBJECT is intentionally used in EVENT, and standardly (for example, in the #$HumanActivitiesMt) OBJECT's role in EVENT is consistent with the object's #$primaryFunction.
> guid: bd5d0531-9c29-11b1-9dad-c379636f7270
> direct instance of: #$ActorSlot
> direct specialization of: #$instrument-Generic

#$hasInterfaceDevices has interface devices **COMMENT NOT REVIEWED** **GAFs NOT REVIEWED**

> This predicate identifies a particular interface device that is linked to a particular computer. (#$hasInterfaceDevices COMPUTER INTERFACE) means that INTERFACE is a #$ComputerInterfaceDevice for the #$Computer COMPUTER. E.g., this predicate holds between my desktop PC and the mouse that's connected to it, the monitor that's connected to it, the keyboard that's connected to it, etc.
> guid: bd58bb61-9c29-11b1-9dad-c379636f7270
> direct instance of: #$AntiSymmetricBinaryPredicate #$PhysicalPartPredicate

#$energySource energy source (binary role predicate) **COMMENT NOT REVIEWED** **GAFs NOT REVIEWED**

> This predicate is used to identify a particular source of energy used in a particular event. (#$energySource EVENT ENERGYSOURCE) means that ENERGYSOURCE provides (some of) the energy used in EVENT. ENERGYSOURCE might be a battery, the sun, a person, etc.
> guid: bd58ea4e-9c29-11b1-9dad-c379636f7270
> direct instance of: #$IndividualLevelPredicate #$BinaryRolePredicate #$AsymmetricBinaryPredicate
> direct specialization of: #$temporallyIntersects

#$objectControlled controlled object **COMMENT NOT REVIEWED** **GAFs NOT REVIEWED**

> This predicate is used to indicate that a particular object is being controlled in a particular event. (#$objectControlled EVENT OBJ) means that the object OBJ is being controlled in the #$Event EVENT. Note: #$objectControlled does not assume or require physical contact between controller and object controlled.
> guid: bd58bdb0-9c29-11b1-9dad-c379636f7270
> direct instance of: #$ActorSlot
> direct specialization of: #$objectActedOn

#$objectActedOn affected object (actor slot)

> The predicate #$objectActedOn is used to relate an event to an entity or entities significantly affected in that event. The entity or entities in question must exist before the event, but may be either destroyed in the event (see the more specific predicate #$inputsDestroyed), or merely affected by it (for example, see the more specific predicates #$damages and #$objectOfStateChange). (#$objectActedOn EVENT OBJECT) means that OBJECT is altered or affected in EVENT, and the change that OBJECT undergoes is central or focal to understanding EVENT. Thus, scissors are _not_ an #$objectActedOn in a #$HairCuttingEvent. The focal change in a haircut is hair getting shorter; thus, hair is a legitimate #$objectActedOn in a #$HairCuttingEvent. The almost microscopic dulling that scissors undergo in a single haircut is a comparatively insignificant change with respect to a single haircut, considered as a #$HairCuttingEvent.
> guid: bd58bae6-9c29-11b1-9dad-c379636f7270
> direct instance of: #$IndividualLevelPredicate #$ActorSlot
> direct specialization of: #$preActors

#$vehicle vehicle (actor slot) **COMMENT NOT REVIEWED** **GAFs NOT REVIEWED**

> (#$vehicle EVENT VEHICLE) means that VEHICLE is a #$TransportationDevice-Vehicle which is both the #$providerOfMotiveForce and the #$transporter in EVENT. If an object is a #$TransportationDevice-Vehicle and plays the role of #$transporter in some moving event, then it generally will play the role of #$vehicle in that event. Examples: a car plays the role of #$vehicle in driving. Note, however, that a bicycle does not play the role of #$vehicle in bike riding since it is not a provider of motive force. A borderline non-example is someone sitting in their car while it's being pulled by a towtruck; their car is not playing the role of #$vehicle in that event.
> guid: bd58a883-9c29-11b1-9dad-c379636f7270
> direct instance of: #$ActorSlot
> direct specialization of: #$deviceUsed #$transporter #$providerOfMotiveForce

#$stuffUsed stuff used **COMMENT NOT REVIEWED** **GAFs NOT REVIEWED**

> The predicate #$stuffUsed relates an event to some tangible substance which facilitates that event. (#$stuffUsed EVENT STUFF) means that STUFF is a portion of an instance of #$ExistingStuffType which plays an instrumental role in EVENT. STUFF may or may not be consumed in the course of EVENT. Examples: portions of #$Water are #$stuffUsed in instances of #$WashingDishes, #$WashingHair, #$WashingClothesInAMachine, etc.; portions of #$EdibleOil are #$stuffUsed in some instances of #$Frying food and #$BakingBread.
> guid: bd58cadc-9c29-11b1-9dad-c379636f7270
> direct instance of: #$ActorSlot
> direct specialization of: #$instrument-Generic

#$transporter transporter (actor slot) **COMMENT NOT REVIEWED** **GAFs NOT REVIEWED**

> (#$transporter MOVE OBJ) means that OBJ enables or facilitates the conveyance of the #$transportees in the #$TransportationEvent MOVE. OBJ is an #$objectMoving in MOVE that moves along with the #$transportees. OBJ will generally hold, support, contain, pull, or push the #$transportees throughout the MOVE #$Event. OBJ may or may not be the #$providerOfMotiveForce in the event MOVE. If OBJ stays with each #$primaryObjectMoving from the #$fromLocation to the #$toLocation, moving along the same trajectory, then it is also a #$primaryObjectMoving. If OBJ facilitates the motion of the #$primaryObjectMoving but does not itself engage in translational motion it is merely the conveyor of the action, and the role #$conveyor-Stationary should be asserted. If it is unclear whether the conveyor is stationary or not, the role #$conveyor-Generic is used. Specializations of the role #$transporter should be used when possible: although automobiles are #$transporters in many events, they should normally have the more specific role designation of #$vehicle because they are also #$SelfPoweredDevices. When a car is being towed by a towtruck, the car is just a #$transporter of any of its contents while the towtruck is the #$vehicle of that event. An additional role designation for some #$transporters -- those which are #$PhysicalDevices -- is #$deviceUsed; e.g., the use of crutches in hobbling or ice skates in skating. See the #$comment on #$TransportationEvent. Note that an organism may be a #$transportees in a #$Bicycle riding or #$Skating event as well as being the #$providerOfMotiveForce in such cases.
> guid: bd58c1d8-9c29-11b1-9dad-c379636f7270
> direct instance of: #$ActorSlot
> direct specialization of: #$conveyor-Generic #$objectMoving

#$transportees transportee (actor slot) **COMMENT NOT REVIEWED** **GAFs NOT REVIEWED**

> The predicate #$transportees relates a translational motion event to the object(s) transported by a separate object, i.e. a distinctly separate other participant in the event. (#$transportees MOVE OBJ) means that some #$conveyor-Generic facilitates the conveyance of OBJ in MOVE. For example, in a dumptruck driving event, the dirt in the back of the truck is a #$transportees. Any humans in the truck cab (or truck bed) during the trip are also #$transportees; however, a more precise role designation for humans riding in the truck would be either #$passengers or (for the driver) #$driverActor. Borderline positive example #$transportees include the clothes worn by a person walking, or a horseshoe worn by a horse walking. A negative exemplar of a #$transportees is the ear of the person walking. This is because #$transporters do not transport their parts when they move. In other words, #$transporters only transport separate objects. #$translatesFromTo on the other hand, does apply to parts of #$transportees. Note also that parts of #$transportees are not necessarily #$transportees themselves. See also the comments on #$TransportationEvent and #$transporter.
> guid: bd58ae94-9c29-11b1-9dad-c379636f7270
> direct instance of: #$ActorSlot
> direct specialization of: #$primaryObjectMoving #$objectActedOn

#$driverActor driver (actor slot) **COMMENT NOT REVIEWED** **GAFs NOT REVIEWED**

> (#$driverActor DRIVE DRIVER) means that DRIVER controls (see #$ControllingATransporter) the #$transporter in DRIVE. DRIVER steers the wheel, grasps the tiller, controls the throttle, the reins, the brakes, etc., of the #$transporter, e.g., a boat, train, windsurfer, mule, plane, horse and carriage, spaceship, sled, etc. DRIVER is not a #$passengers in DRIVE. Because #$transporter and #$transportees are disjoint and #$driverActor has #$transportees as a #$genlPreds, DRIVER is distinct from the value on #$transporter. Thus a person walking while carrying a watermelon would not be a #$driverActor in their own walking. DRIVER is usually in #$SittingPosture during DRIVE. For any given instant of DRIVE there is exactly one DRIVER. Until we have more extensive vocabulary, #$transportees is the most specific we can be about events in which multiple drivers share driving responsibility throughout the event or if there is a pilot/co-pilot combination.
> guid: bd58bf45-9c29-11b1-9dad-c379636f7270
> direct instance of: #$AgentiveRole
> direct specialization of: #$crewMember #$performedBy #$transportees

### Vectors

#$VectorInterval relative location **COMMENT NOT REVIEWED** **GAFs NOT REVIEWED**

> A specialization of #$NTupleInterval. Each instance of #$VectorInterval is an n-tuple of intervals (where n > 1), one of which is a direction. Like the instances of #$ScalarInterval, the intervals in an instance of #$VectorInterval may be point-valued or cover a range of values. The minimal interval (i.e., point-valued) type of vector interval is exemplified by a vector such as `10 meters due east'. Vectors may also cover a range of values; e.g., `at least 10 feet away and in a horizontal direction'; `between ten to twelve miles NNW'.
> guid: bd58ec12-9c29-11b1-9dad-c379636f7270
> direct instance of: #$ObjectType
> direct specialization of: #$NTupleInterval
> direct generalization of: #$UnitVectorInterval #$Vector-Precise

#$Vector-Precise precise vectors **COMMENT NOT REVIEWED** **GAFs NOT REVIEWED**

> A specialization of #$VectorInterval. Each instance of #$Vector-Precise is an exactly indicated (i.e., point) vector, such as `5 feet due West'. Both direction and distance are precise. Thus, #$Vector-Precise is to #$VectorInterval what #$ScalarPointValue is to #$ScalarInterval. #$Vector-Precise includes all the instances of #$UnitVector-Precise.
> guid: bd58dd86-9c29-11b1-9dad-c379636f7270
> direct instance of: #$ObjectType
> direct specialization of: #$VectorInterval
> direct generalization of: #$UnitVector-Precise

#$UnitVectorInterval unit vector intervals **COMMENT NOT REVIEWED** **GAFs NOT REVIEWED**

> The collection #$UnitVectorInterval is a subcollection of #$VectorInterval. Each instance of #$UnitVectorInterval is a vector interval with a magnitude of 1. The range of the endpoints of all the unit vectors [#$UnitVector-Precise] form a contiguous curve (in 2 space), surface (in 3 space), volume (in 4 space), etc. depending upon the dimentionality of the vector. One #$UnitVectorInterval differs from another only in range of direction, since the magnitude of every #$UnitVectorInterval is the same. An instance of #$UnitVectorInterval may specify either a precise unit vector [#$UnitVector-Precise] or a generalized range of directions such as `in front of'. The range of directions in 3-space may be quite irregular, such as the direction interval from #$Chile to #$Russia.
> guid: bd58eb0c-9c29-11b1-9dad-c379636f7270
> direct instance of: #$ObjectType
> direct specialization of: #$VectorInterval
> direct generalization of: #$TerrestrialDirection #$UnitVector-Precise #$DirectionExpression

#$UnitVector-Precise unit vector - precise **COMMENT NOT REVIEWED** **GAFs NOT REVIEWED**

> A specialization of both #$Vector-Precise and #$UnitVectorInterval. Each instance of #$UnitVector-Precise is a vector interval with a magnitude of 1 and a precisely specified direction (e.g., due North, straight down). Thus, one precise unit vector differs from another only in direction, since each vector consists of a magnitude and a direction (in a space of n > 1 dimensions). So instances of #$UnitVector-Precise in effect indicate different directions such as `straight up' or `due East'.
> guid: bd58ec98-9c29-11b1-9dad-c379636f7270
> direct instance of: #$ObjectType
> direct specialization of: #$UnitVectorInterval #$Vector-Precise
> direct generalization of: #$GeographicalDirection-Direct

### Subsets Of Mathematical Thing

#$MathematicalThing mathematical concepts **COMMENT NOT REVIEWED** **GAFs NOT REVIEWED**

> A specialization of #$MathematicalOrComputationalThing. Each instance of #$MathematicalThing is an atemporal, nonspatial, purely mathematical thing. #$MathematicalThing is partitioned into two main specializations, #$MathematicalObject and #$SetOrCollection (qq.v).
> guid: bd58e5b6-9c29-11b1-9dad-c379636f7270
> direct instance of: #$VariableOrderCollection
> direct specialization of: #$MathematicalOrComputationalThing
> direct generalization of: #$MathematicalObject #$SetOrCollection

#$MathematicalObject mathematical object **COMMENT NOT REVIEWED** **GAFs NOT REVIEWED**

> A specialization of both #$MathematicalThing and #$IntangibleIndividual. Each instance of #$MathematicalObject is a purely abstract mathematical thing which is also an individual (see #$Individual). Specializations of #$MathematicalObject include #$Quantifier, #$RealNumber, #$Triangle, and #$TruthValue. Note that instances of #$SetOrCollection are not instances of #$MathematicalObject, since they are not instances of #$Individual.
> guid: bf461f37-9c29-11b1-9dad-c379636f7270
> direct instance of: #$ObjectType
> direct specialization of: #$MathematicalThing #$IntangibleIndividual
> direct generalization of: #$Relation-MathematicalObject #$RelationalStructure #$Number-General #$GeometricThing-Abstract #$FrameOfReference #$Relation #$Tuple #$TruthValue

#$TruthValue truths **COMMENT NOT REVIEWED** **GAFs NOT REVIEWED**

> #$TruthValue is a collection of mathematical objects; it contains the abstract, logical objects #$True and #$False.
> guid: bd58e3ba-9c29-11b1-9dad-c379636f7270
> direct instance of: #$ObjectType
> direct specialization of: #$MathematicalObject

#$True true (mathematical concept) **COMMENT NOT REVIEWED** **GAFs NOT REVIEWED**

> An instance of #$TruthValue. #$True is logical truth in Cyc; this is the abstract logical notion--not to be confused with Lisp's T, nor with the English word `true'.
> guid: bd5880d9-9c29-11b1-9dad-c379636f7270
> direct instance of: #$TruthValue #$Individual

#$False false (mathematical concept) **COMMENT NOT REVIEWED** **GAFs NOT REVIEWED**

> An instance of #$TruthValue. #$False is logical falsehood in Cyc; this is the abstract logical notion--not to be confused with Lisp's NIL, nor with the English word `false'.
> guid: bd5880d8-9c29-11b1-9dad-c379636f7270
> direct instance of: #$TruthValue #$Individual

#$Relation relationships (mathematical concepts)

> The collection of relations whose CycL representations can appear in the 0th (or arg0 ) argument position of a #$CycLFormula, i.e. as the term immediately following the formula's opening parenthesis. An important subcollection of #$Relation is #$TruthFunction (q.v.), whose instances are intimately related to truth-values, as reflected in the fact that the CycL expressions that denote truth-functions can appear in the arg0 position of a #$CycLSentence; and a sentence (if quantificationally closed; see #$CycLClosedSentence), will generally be either true or false (with respect to a given context or interpretation). The major subcollections of #$TruthFunction are #$Predicate, #$LogicalConnective, and #$Quantifier (qq.v.). Another important subcollection of #$Relation is #$Function-Denotational (q.v.), whose instances are functions the CycL expressions for which can appear in the arg0 position of a #$CycLNonAtomicTerm; and such terms (if closed) generally denote things.
> guid: bd5880cd-9c29-11b1-9dad-c379636f7270
> direct instance of: #$RelationshipType
> direct specialization of: #$MathematicalObject
> direct generalization of: #$MicrotheoryDesignatingRelation #$PartiallyCommutativeRelation #$AssociativeRelation #$ScopingRelation #$EvaluatableRelation #$CommutativeRelation #$FunctionalRelation #$MacroRelation #$FixedArityRelation #$VariableArityRelation #$Function-Denotational #$TruthFunction

#$Quantifier quantifier (relationship) **COMMENT NOT REVIEWED** **GAFs NOT REVIEWED**

> A collection of mathematical objects. Each instance of #$Quantifier represents a relationship between a variable and a formula. In Cyc, a quantifier binds the variable found in its first argument within the formula that appears as its second argument. Instances of #$Quantifier in CycL include #$forAll, #$thereExists, #$thereExistExactly, #$thereExistAtLeast, #$thereExistAtMost.
> guid: bd58c271-9c29-11b1-9dad-c379636f7270
> direct instance of: #$RelationshipType
> direct specialization of: #$ScopingRelation #$TruthFunction
> direct generalization of: #$ExistentialQuantifier

#$LogicalConnective logical connective **COMMENT NOT REVIEWED** **GAFs NOT REVIEWED**

> A collection of mathematical objects, including the basic logical connectives. Each instance of #$LogicalConnective is a #$Relation which takes one or more truth-valued expressions (sentences) as arguments and returns a truth-valued sentence. The instances of #$LogicalConnective include #$and, #$or, #$not, and #$implies.
> guid: bd58b9f9-9c29-11b1-9dad-c379636f7270
> direct instance of: #$RelationshipType
> direct specialization of: #$TruthFunction

#$VariableArityRelation variable arity relation

> A specialization of #$Relation. Each instance of #$VariableArityRelation is a relation that can take a variable number of arguments. Examples of #$VariableArityRelation include the predicate #$different and the function #$PlusFn.
> guid: bd5d0b56-9c29-11b1-9dad-c379636f7270
> direct instance of: #$RelationshipTypeByArity
> direct specialization of: #$Relation
> direct generalization of: #$SocialOrEconomicAttributeFunction #$VariableArityFunction #$VariableAritySkolemFunction

#$Tuple tuple **GAFs NOT REVIEWED**

> A collection of mathematical objects. Each instance of #$Tuple is a complex consisting of one or more ordered (or otherwise indexed) components; it might be a pair, or a triple, or so on; and the components might be things of any sort whatsoever. For example, #$BloodPressureReading is a specialization #$Tuple, each instance of which is an ordered or column-indexed pair of numbers, where the first number is the systolic reading and the second the diastolic reading. Another specialization of #$Tuple is #$NTupleInterval (q.v.), whose instances are tuples consisting exclusively of #$ScalarIntervals (q.v.); e.g. complex numbers and physical vectors are n-tuple intervals. If the index set for a given #$Tuple is the counting numbers (or an initial segment of them), then the numbers' usual ordering serves to order the tuple's components, and the tuple is in fact a #$List (q.v.). But in general any set (e.g. the column names in a relational database) may be used to index the components of a tuple.
> guid: bd58f8d8-9c29-11b1-9dad-c379636f7270
> direct instance of: #$ObjectType
> direct specialization of: #$MathematicalObject
> direct generalization of: #$NTupleInterval #$List

#$Fractal fractal (mathematical concept) **COMMENT NOT REVIEWED** **GAFs NOT REVIEWED**

> A collection of functions. Each element of #$Fractal is a function which, when applied to data, can be displayed visually as an instance of #$FractalRepresentation.
> guid: c10c1ae4-9c29-11b1-9dad-c379636f7270
> direct instance of: #$ObjectType
> direct specialization of: #$Function-MathematicalObject

#$GeometricallyDescribableThing geometric forms **GAFs NOT REVIEWED**

> A subcollection of #$SpatialThing. Each instance of #$GeometricallyDescribableThing is a spatially-connected spatial thing (of 0, 1, 2, or 3 dimensions) that either (i) has or approximates a simple geometric shape (e.g. it is a #$Line of a #$Hemisphere) or (ii) consists of a number of (connected) parts in a relatively stable geometric configuration, where each such part has or approximates a simple geometric shape (e.g. a table consisting of a 3-D-disc-shaped top and four cylindrical legs). A #$GeometricallyDescribableThing might be tangible (see #$PartiallyTangible) or intangible (see #$GeometricallyDescribableThing-Intangible). Note that what counts as approximating a given simple geometric shape -- and thus what spatial things count as #$GeometricallyDescribableThings -- varies with context. In a context that was so fine-grained shape-wise that even the shapes of the individual molecules on the surface of an object were considered relevant to the object's shape, perhaps nearly every (connected, solid) tangible object would be geometrically-describable. In more everyday contexts, on the other hand, an unopened can of soup would be geometrically-describable (as a cylinder), while a telephone or an animal's body would probably not.
> guid: bd58c42e-9c29-11b1-9dad-c379636f7270
> direct instance of: #$ObjectType
> direct specialization of: #$SpatialThing
> direct generalization of: #$GeometricallyDescribableThing-Intangible #$GeometricThing-Localized #$LinePortion #$HalfLine-Geometrical #$RoundShape #$Angle #$Line #$TwoDimensionalGeometricThing #$ThreeDimensionalGeometricThing

#$GeometricThing-Abstract abstract shapes **COMMENT NOT REVIEWED** **GAFs NOT REVIEWED**

> A specialization of #$GeometricallyDescribableThing each of whose instances is abstract in the sense of being intangible (see #$Intangible) as well as lacking spatial and temporal location. Each instance of #$GeometricThing-Abstract is an abstract region of an abstract space (the latter having two or more dimensions). Geometric figures that are located in this (or another) universe are not instances of this collection, but of #$GeometricThing-Localized.
> guid: bd5885bc-9c29-11b1-9dad-c379636f7270
> direct instance of: #$ObjectType
> direct specialization of: #$GeometricallyDescribableThing-Intangible #$MathematicalObject

#$ThreeDimensionalGeometricThing three dimensional shapes **COMMENT NOT REVIEWED** **GAFs NOT REVIEWED**

> A specialization of #$GeometricallyDescribableThing. Each instance of #$ThreeDimensionalGeometricThing is a three-dimensional object. Examples include spatially localized objects, such as the Pentagon, as well as abstract three-dimensional shapes.
> guid: c0fbbe61-9c29-11b1-9dad-c379636f7270
> direct instance of: #$ThreeDimensionalShapeType
> direct specialization of: #$GeometricallyDescribableThing #$TwoOrHigherDimensionalThing
> direct generalization of: #$Cruciform #$SpiralShaped #$Screw-GenericShape #$Rectangular3DShape #$Hemisphere #$CylinderShape #$Polygon3DShaped #$Ellipsoid #$SphereShape #$ConeShape #$CubeShape #$PyramidShape #$KnotShape

#$FrameOfReference frames of reference **COMMENT NOT REVIEWED** **GAFs NOT REVIEWED**

> A specialization of #$MathematicalObject. Each instance of #$FrameOfReference is a mathematical (and hence intangible) representation of the context in which certain data are to be interpreted. Such contexts are typically physical (i.e., spatiotemporal), but contexts may also be purely mathematical. A Cartesian coordinate system represents a frame of reference.
> guid: bd58d4a0-9c29-11b1-9dad-c379636f7270
> direct instance of: #$ObjectType
> direct specialization of: #$MathematicalObject

#$Line lines **COMMENT NOT REVIEWED** **GAFs NOT REVIEWED**

> A specialization of #$GeometricallyDescribableThing. Each instance of #$Line is a one-dimensional path, either curved or straight, through one-, two-, or higher-dimensional space. Examples include spatially localized objects, such as the equator, as well as abstract lines.
> guid: bd5906cb-9c29-11b1-9dad-c379636f7270
> direct instance of: #$GenericShapeType
> direct specialization of: #$Linear #$GeometricallyDescribableThing
> direct generalization of: #$Arc #$Line-Straight

#$TwoDimensionalGeometricThing two dimensional shapes **COMMENT NOT REVIEWED** **GAFs NOT REVIEWED**

> The collection of #$GeometricallyDescribableThings (q.v.) that are two-dimensional. Each instance of #$TwoDimensionalGeometricThing is a two-dimensional object whose shape is describable in geometric terms. Examples include tangible objects, such as the flat (two-dimensional) tangible surface of an oval tabletop, as well as abstract two-dimensional objects.
> guid: bd58c2a5-9c29-11b1-9dad-c379636f7270
> direct instance of: #$GenericShapeType #$TwoDimensionalShapeType
> direct specialization of: #$GeometricallyDescribableThing #$TwoOrHigherDimensionalThing
> direct generalization of: #$SectorShaped #$Oval #$PlaneFigure-Geometrical #$HalfPlane #$Quadrilateral #$Ellipse #$Polygon #$Circle

#$Angle angles **COMMENT NOT REVIEWED** **GAFs NOT REVIEWED**

> A specialization of #$GeometricallyDescribableThing. Each instance of #$Angle is formed by two lines diverging from the same point, or two surfaces diverging from the same line. Examples include spatially localized objects, such as the angle formed by the intersection of two walls, and abstract objects, such as the angle formed by the intersection of two (abstract) lines.
> guid: bd61bd87-9c29-11b1-9dad-c379636f7270
> direct instance of: #$GenericShapeType
> direct specialization of: #$GeometricallyDescribableThing

#$NTupleInterval n tuple interval **COMMENT NOT REVIEWED** **GAFs NOT REVIEWED**

> A specialization of #$Tuple. Each instance of #$NTupleInterval is a tuple whose items are all intervals. Notable specializations of #$NTupleInterval include #$ScalarInterval, #$VectorInterval, and #$ComplexNumber.
> guid: bd58ec55-9c29-11b1-9dad-c379636f7270
> direct instance of: #$ObjectType
> direct specialization of: #$Tuple
> direct generalization of: #$VectorInterval #$ScalarInterval #$ComplexNumber

#$ScalarInterval quantities **COMMENT NOT REVIEWED** **GAFs NOT REVIEWED**

> A specialization of #$NTupleInterval. Each proper subcollection SCALAR of #$ScalarInterval (with some exceptions, such as #$ScalarPointValue) has the following two properties: 1) the collection of point instances of SCALAR (i.e., those instances of SCALAR that are _not_ proper intervals - see the collection #$ScalarPointValue) is ordered by some `natural' linear ordering (i.e., some `natural' relation that is reflexive, antisymmetric, and transitive on the collection of point instances of SCALAR), 2) SCALAR is closed under addition (#$PlusFn), so that the sum of any two instances of SCALAR will also be an instance of SCALAR. Notable specializations of #$ScalarInterval include #$Time-Quantity, #$Integer, and #$Distance.
> guid: bd5880a7-9c29-11b1-9dad-c379636f7270
> direct instance of: #$ObjectType
> direct specialization of: #$NTupleInterval
> direct generalization of: #$PhysicalQuantity #$NegativeScalarInterval #$JustificationStrength #$TemperamentAttribute #$Interval-UnboundedAbove #$Interval-BoundedBelow #$Interval-UnboundedBelow #$Interval-BoundedAbove #$NonNegativeScalarInterval #$LevelOfDiscomfort #$LevelOfPain #$OrderOfMagnitudeInterval #$ScalarPointValue #$IntervalOnNumberLine #$NonPositiveScalarInterval #$Dirtiness #$Money #$PolitenessOfSpeech #$FormalityOfSpeech #$FeelingAttribute #$Alertness

#$ScalarPointValue scalar values **COMMENT NOT REVIEWED** **GAFs NOT REVIEWED**

> The collection of minimal scalar intervals. That is, each instance of #$ScalarPointValue is a scalar interval which has no `width'. Scalar intervals include both pure numbers, such as 3.14, and linear measurements such as 3.14 meters, which in CycL would be written `(#$Meter 3.14)'.
> guid: bd58825b-9c29-11b1-9dad-c379636f7270
> direct instance of: #$ObjectType
> direct specialization of: #$Interval-Bounded
> direct generalization of: #$RealNumber #$Cardinal-Mathematical

#$ComplexNumber complex number **COMMENT NOT REVIEWED** **GAFs NOT REVIEWED**

> A specialization of both #$Number-General and #$NTupleInterval. Each instance of #$ComplexNumber can be thought of as a vector of two numbers, which are usually called the real part and the imaginary part of the complex number. Complex numbers may also be considered as corresponding to points in the real plane, where the x axis determines the real component of a complex number and the y axis the imaginary component. The unit value on the real number line is 1, the unit value on the imaginary number line is the square root of -1, generally written `i' in mathematics and `j' in engineering.
> guid: bd58b80a-9c29-11b1-9dad-c379636f7270
> direct instance of: #$ObjectType
> direct specialization of: #$Number-General
> direct generalization of: #$RealNumber

#$MathematicalPoint mathematical point **COMMENT NOT REVIEWED** **GAFs NOT REVIEWED**

> A collection of mathematical objects. Each element of #$MathematicalPoint is an atomic (i.e., structureless) abstract object.
> guid: bd5903d9-9c29-11b1-9dad-c379636f7270
> direct instance of: #$ObjectType
> direct specialization of: #$MathematicalObject

#$Field-Mathematical fields **COMMENT NOT REVIEWED** **GAFs NOT REVIEWED**

> A collection of spatial things. Each element of #$Field-Mathematical is an abstract region through which forces (or other vector, scalar, or tensor functions of position) are exerted; often, relative to one or two objects.
> guid: bd58eea6-9c29-11b1-9dad-c379636f7270
> direct instance of: #$Collection
> direct specialization of: #$SpatialThing #$IntangibleIndividual

#$SetOrCollection intensional or extensional sets

> A specialization of #$MathematicalThing. Something is an instance of #$SetOrCollection just in case it is a collection (i.e. an instance of #$Collection) or a mathematical set (i.e. an instance of #$Set-Mathematical). Instances of #$Set-Mathematical and instances of #$Collection (and thus instances of #$SetOrCollection) share some basic common features. All instances of #$Collection and all instances of #$Set-Mathematical (and thus all instances of #$SetOrCollection) are abstract entities, lacking spatial and temporal properties. Nearly all instances of #$Collection (except empty collections) and nearly all instances of #$Set-Mathematical (except the empty set; see #$TheEmptySet) have elements (i.e. instances or members; see #$elementOf); hence set-or-collections may stand to one another in generalized set-theoretic relations such as #$subsetOf and #$disjointWith (qq.v.). (It is this shared feature of having elements that provides the basic rationale for reifying the collection #$SetOrCollection.) Nevertheless, sets and collections differ in two important ways. First, each collection is intrinsically associated with an intensional criterion for membership -- a more or less natural property (or group of properties) possessed by all of (and only) its elements. Collections are thus akin to kinds. In contrast, the elements of a set are not required to be homogeneous in any respect: any things whatsoever may together constitute the elements of a set. The second major difference between sets and collections is that no two distinct sets can be coextensional (i.e. have exactly the same elements; see #$coExtensional). Sets can thus be identified purely on the basis of their extensions (see #$extent). Collections, on the other hand, are individuated by their intensional criteria for membership. So collections that have exactly the same elements might nevertheless be distinct, differing in their respective membership criteria. (Note that the general relationship between collections and their intensional criteria for membership in the above sense is not something that is currently represented explicitly in the Knowledge Base (though this seems a worthwhile area for future work); still the #$comment and other definitional assertions on a given collection should ideally convey a reasonably clear and precise idea of its associated membership criterion.)
> guid: bd58e5fd-9c29-11b1-9dad-c379636f7270
> direct instance of: #$VariableOrderCollection #$SetOrCollectionType
> direct specialization of: #$MathematicalThing
> direct generalization of: #$SetOrCollectionType #$Set-Mathematical #$Collection

#$Set-Mathematical sets (mathematical concepts) **COMMENT NOT REVIEWED** **GAFs NOT REVIEWED**

> The collection of mathematical sets. An element of #$Set-Mathematical can be any arbitrary set, including sets whose members have nothing in common. In contrast, the members of an instance of #$Set-Mathematical's sibling #$Collection (q.v.) all have some important, natural properties in common. Sets and collections also differ in that there cannot exist two distinct sets that have exactly the same elements. A third point of contrast between sets and collections is that rarely will it be desirable to create a new constant to refer to a set. Instead, a set will either be intensionally specified by a defining property, using #$TheSetOf, as in (#$TheSetOf ?X (#$and (#$isa ?X #$PositiveInteger)(#$greaterThan ?X 42))), or extensionally specified by listing its elements, using #$TheSet, as in (#$TheSet 3 4 5). (In certain cases, a set will be extensionally specified by means of one of the more specialized functions #$ThePartition or #$TheCovering. See #$partitionedInto and #$covering.)
> guid: be5d9e9f-9c29-11b1-9dad-c379636f7270
> direct instance of: #$Collection
> direct specialization of: #$SetOrCollection

#$DisjointSetOrCollection disjoint set or collections

> A collection of mathematical sets and collections whose elements are themselves mathematical sets or collections. A set or collection, SETORCOL, of sets or collections is an instance of #$DisjointSetOrCollection just in case the elements of SETORCOL are mutually disjoint -- that is, no two elements of SETORCOL have any elements in common.
> guid: be13fa12-9c29-11b1-9dad-c379636f7270
> direct instance of: #$VariableOrderCollection #$SetOrCollectionType
> direct specialization of: #$SiblingDisjointSetOrCollection
> direct generalization of: #$DisjointCollectionType

#$subsetOf subset (taxonomic slot)

> This predicate relates a set or collection SUB to a set or collection SUPER whenever the extent (see #$extent) of SUB is a subset of the extent of SUPER. That is, (#$subsetOf SUB SUPER) means that every element of (see #$elementOf) SUB is an element of SUPER. #$subsetOf is thus a generalization both of the subset relation in set theory and of #$genls (q.v.); and (unlike either of those other relations) #$subsetOf can hold between a set and a collection, or between a collection and a set.
> guid: bd903ed3-9c29-11b1-9dad-c379636f7270
> direct instance of: #$TaxonomicSlot #$ReflexiveBinaryPredicate #$TransitiveBinaryPredicate
> direct specialization of: #$most-GenQuant #$generalizations

#$elementOf element of

> (#$elementOf THING SETORCOL) means that THING is an element of the mathematical set or collection SETORCOL. #$elementOf is a more general relation than #$isa, since #$isa is used exclusively to talk about membership in instances of #$Collection. #$elementOf, unlike #$isa, can also be used to talk about membership in arbitrarily-defined mathematical sets (instances of #$Set-Mathematical), such as those denoted by #$TheSet expressions.
> guid: c0659a2b-9c29-11b1-9dad-c379636f7270
> direct instance of: #$TaxonomicSlotForAnyThing

#$TheSetOf the set of **COMMENT NOT REVIEWED** **GAFs NOT REVIEWED**

> A function that results in a #$Set-Mathematical of instances that satisfy some #$ELSentence-Assertible. The first argument is an #$ELVariable that appears as a free variable in the second argument. For example, (#$TheSetOf ?X (#$and (#$isa ?X #$Dog) (#$objectHasColor ?X #$BlackColor) denotes the set of black dogs.
> guid: bd58d7f6-9c29-11b1-9dad-c379636f7270
> direct instance of: #$UnreifiableFunction #$BinaryFunction #$ScopingRelation #$SetDenotingFunction #$Individual

#$TheSet the set (relationship) **COMMENT NOT REVIEWED** **GAFs NOT REVIEWED**

> (#$TheSet E1 E2 ... En) denotes the #$Set-Mathematical consisting of the elements E1 through En. #$TheSet is a variable arity relation, taking one or more arguments. All the arguments must be ground terms; variables are not allowed. See also #$TheSetOf to define sets in the {x: ---x---} format.
> guid: bd58e476-9c29-11b1-9dad-c379636f7270
> direct instance of: #$VariableArityFunction #$UnreifiableFunction #$CommutativeRelation #$SetDenotingFunction #$Individual

#$TermSet term set **COMMENT NOT REVIEWED** **GAFs NOT REVIEWED**

> A collection of sets. Each element of #$TermSet is a set of Cyc terms.
> guid: bd597688-9c29-11b1-9dad-c379636f7270
> direct instance of: #$ObjectType
> direct specialization of: #$Set-Mathematical

### Relational Structures

#$RelationalStructure relational structure **COMMENT NOT REVIEWED** **GAFs NOT REVIEWED**

> The collection of all mathematical structures each being a composite individual structure consisting of a #$baseSet with structuring relations or operations on that set (and, optionally, one or more selected other sets, relations, functions or individuals). Examples include #$PartialOrderings, #$Multigraphs, etc. Sometimes mathematicians specify these using a #$Tuple of sets, relations, functions, and/or individuals, as specifications. Note that RelationalStructure is not the same as its #$baseSet since a pure #$SetOrCollection necessarily lacks the associated 'structure'.
> guid: bf48bfa0-9c29-11b1-9dad-c379636f7270
> direct instance of: #$ObjectType
> direct specialization of: #$MathematicalObject
> direct generalization of: #$MathematicalOrdering #$Multigraph

#$baseSet base set **COMMENT NOT REVIEWED** **GAFs NOT REVIEWED**

> (#$baseSet STRUCTURE SET) means that SET is the base set of #$RelationalStructure STRUCTURE. That is, SET is the domain from which the individual elements of the structure are drawn. Although there may be several relations involved in the relational structure, they all relate members of the base set. As each relational structure has a unique base set, this predicate is functional.
> guid: be51a217-9c29-11b1-9dad-c379636f7270
> direct instance of: #$FunctionalSlot #$IntangibleObjectRelatingPredicate

#$orderingRelations ordering relations (asymmetric binary predicate) **COMMENT NOT REVIEWED** **GAFs NOT REVIEWED**

> (#$orderingRelations ORDER PRED) means that, in a #$MathematicalOrdering ORDER, there is an order-predicate PRED that forms an ordering relation on the #$baseSet of ORDER. The predicate PRED, when restricted to the #$baseSet of the #$MathematicalOrdering ORDER, is transitive on the #$baseSet of ORDER. Note that the predicate #$orderingRelations is not a functional predicate because we may have both (#$orderingRelations ORDER PRED1) and (#$orderingRelations ORDER PRED2). (This is true even if, extensionally, the two predicates PRED1 and PRED2 denote only one set-theoretic binary relation associated with ORDER.) Such an ordering relation is not necessarily a full order or even a partial order; it depends on the kind of #$MathematicalOrdering.
> guid: c0416aa0-9c29-11b1-9dad-c379636f7270
> direct instance of: #$AsymmetricBinaryPredicate

#$MathematicalOrdering mathematical ordering **COMMENT NOT REVIEWED** **GAFs NOT REVIEWED**

> The collection of all those #$RelationalStructures that are called 'orderings'. A #$MathematicalOrdering is usually described as an ordered pair <S, R> where S is a set and R is a binary relation on S that is transitive, i.e., for each X, Y and Z in S, R(X Y) and R(Y Z) imply R(X Z). We do not have to define a #$MathematicalOrdering in Cyc as an ordered pair, but it is essential that each such #$MathematicalOrdering has a unique #$baseSet and a unique ordering relation. We use, for each #$MathematicalOrdering ORDER, (#$orderingRelations ORDER PRED) to specify a binary predicate PRED, the restriction of which to the #$baseSet S of ORDER indicates the ordering relation R on S, and we require the collections that are used to specify the arguments to PRED to be supersets of S. In principle, there could be different predicates PRED1 and PRED2 such that when restricted to the same #$baseSet S of a #$MathematicalOrdering ORDER, they order the elements of S exactly the same way. When we said that there is a unique ordering relation R on S, we mean to ignore the difference between PRED1 and PRED2 when they are restricted to S, and treat the results of such restrictions the same, as far as they are used to talk about ORDER. Subcollections of #$MathematicalOrdering include #$PartialOrdering, #$PartialOrdering-Strict, #$TreeOrdering, #$TreeOrdering-Strict, #$TotalOrdering, #$TotalOrdering-Strict, #$WellOrdering and #$Lattice-LatticeTheoretic, etc.
> guid: bf3b5382-9c29-11b1-9dad-c379636f7270
> direct instance of: #$ObjectType
> direct specialization of: #$RelationalStructure
> direct generalization of: #$QuasiOrdering #$PartialOrdering-Strict

#$QuasiOrdering quasi-orderings **COMMENT NOT REVIEWED** **GAFs NOT REVIEWED**

> The collection of all those #$MathematicalOrderings ORDER in which the ordering relation R is a reflexive and transitive relation on the #$baseSet S of ORDER, i.e., for each X in S, R(X X) holds, and for each X, Y and Z in S, if R(X Y) and R(Y Z) then R(X Z). For example, if you take the set of all people in the states today, and take the relation '__ is at least as tall as ...' (i.e., either __ is as tall as ... or __ is taller than ...) on this set, you get a #$QuasiOrdering because this relation is reflexive and transitive on the set of all people in the states today. Note that the #$QuasiOrdering in this example is neither a #$PartialOrdering nor a #$PartialOrdering-Strict. Subcollections of #$QuasiOrdering include #$PartialOrdering, #$TreeOrdering, #$TotalOrdering and #$Lattice-LatticeTheoretic.
> guid: c14247c0-9c29-11b1-9dad-c379636f7270
> direct instance of: #$ObjectType
> direct specialization of: #$MathematicalOrdering
> direct generalization of: #$PartialOrdering

#$PartialOrdering partial ordering **COMMENT NOT REVIEWED** **GAFs NOT REVIEWED**

> The collection of all those #$MathematicalOrderings ORDER in which the ordering relation R is a reflexive, transitive and antisymmetric relation on the #$baseSet S of ORDER. R is reflexive on S if and only if for each X in S, R(X X). R is transitive on S if and only if for each X, Y and Z in S, R(X Y) and R(Y Z) imply R(X Z). R is antisymmetric on S if and only if for each X and Y in S, R(X Y) and R(Y X) imply X = Y. For example, if you take a set of #$Lists and take the #$subLists relation restricted to this set, then you have a #$PartialOrdering because #$subLists relation is reflexive, transitive and antisymmetric. Since the ordering relation in each #$PartialOrderings reflexive and transitive, the collection #$PartialOrdering is a subcollection of #$QuasiOrdering. Subcollections of #$PartialOrdering include #$TreeOrdering, #$TotalOrdering and #$Lattice-LatticeTheoretic. If you want a #$MathematicalOrdering in which the ordering relation is irreflexive, see #$PartialOrdering-Strict.
> guid: c135aad5-9c29-11b1-9dad-c379636f7270
> direct instance of: #$ObjectType
> direct specialization of: #$QuasiOrdering
> direct generalization of: #$TreeOrdering #$Lattice-LatticeTheoretic

#$Lattice-LatticeTheoretic lattice order **COMMENT NOT REVIEWED** **GAFs NOT REVIEWED**

> The collection of all mathmetical structures called 'lattices' in Lattice Theory (this is not the same concept as the crystalline or grid lattices studied in Crystallography and Group Theory). A #$Lattice-LatticeTheoretic is often defined in different but equivalent ways. To define a lattice using ordering relation, it is a #$PartialOrdering ORDER in which each pair of elements of the #$baseSet S of ORDER has an R-smallest upper bound and an R-greatest lower bound, where R is the ordering relation on S. To define a lattice using operations, it is a mathematical structure with two operations MEET and JOIN on the #$baseSet S of the structure that satisfy the commutative laws, the associative laws, the idempotent laws and the absorption laws. Note that the correspondence of these two different ways of defining lattices is characterized by the following: for all X, Y in S, Y = (JOIN X Y) <=> R(X Y) <=> X = (MEET X Y). See #$meetFunctionOnLattice and #$joinFunctionOnLattice.
> guid: bdfb75ea-9c29-11b1-9dad-c379636f7270
> direct instance of: #$ObjectType
> direct specialization of: #$PartialOrdering
> direct generalization of: #$ModularLattice

#$ModularLattice modular lattices **COMMENT NOT REVIEWED** **GAFs NOT REVIEWED**

> The subcollection of #$Lattice-LatticeTheoretic that contains all those lattices called 'modular lattices' by the lattice theorists. A #$Lattice-LatticeTheoretic LATTICE (with the #$baseSet S, ordering relation R, meet operation MEET and join operation JOIN) is a #$ModularLattice if the following condition holds: for each X, Y and Z in S, if R(Z X) then (MEET X (JOIN Y Z)) = (JOIN (MEET X Y) (MEET X Z)). Note that there are many conditions equivalent to the one above, one of which is the condition that for each X, Y and Z in S, if R(Z X) then (JOIN X (MEET Y Z)) = (MEET (JOIN X Y) (MEET X Z)). (Other equivalent conditions may be found by browsing the rules for #$ModularLattice.) Note also that a #$DistributiveLattice satisfies this condition automatically, and therefore is a #$ModularLattice.
> guid: c0caf649-9c29-11b1-9dad-c379636f7270
> direct instance of: #$ObjectType
> direct specialization of: #$Lattice-LatticeTheoretic
> direct generalization of: #$DistributiveLattice

#$DistributiveLattice distributive lattices **COMMENT NOT REVIEWED** **GAFs NOT REVIEWED**

> The subcollection of #$Lattice-LatticeTheoretic that contains all those lattices called 'distributive lattices' by lattice theorists. A #$Lattice-LatticeTheoretic LATICE (with the #$baseSet S, meet operation MEET and join operation JOIN) is distributive if the distribution laws hold, i.e., for each X, Y and Z in S, (MEET X (JOIN Y Z)) = (JOIN (MEET X Y) (MEET X Z)). Note that this condition is equivalent to the condition that for each X, Y and Z in S, (JOIN X (MEET Y Z)) = (MEET (JOIN X Y) (JOIN X Z)).
> guid: bdbc5d4c-9c29-11b1-9dad-c379636f7270
> direct instance of: #$ObjectType
> direct specialization of: #$ModularLattice
> direct generalization of: #$TotalOrdering

#$PartialOrdering-Strict strict partial order

> The collection of all #$MathematicalOrderings ORDER in which the ordering relation R is an irreflexive and transitive relation on the #$baseSet S of ORDER, i.e., for each X in S, R(X X) does not hold, and for each X, Y and Z in S, R(X Y) and R(Y Z) imply R(X Z). For example, if one takes the set of all people, and the relation '__ is older than ...' on this set, one gets a #$PartialOrdering-Strict, since the relation '__ is older than ...' is irreflexive and transitive on this set. Note that the important difference between a #$PartialOrdering (q.v.) and a #$PartialOrdering-Strict is that the ordering relation of the former is reflexive, while that of the latter is irreflexive. Note also that the ordering relation R of a #$PartialOrdering-Strict ORDER is in fact antisymmetric (i.e., for each X and Y in S, R(X Y) and R(Y X) imply X = Y) and asymmetric (i.e., for each X and Y in S, R(X Y) and R(Y X) can never be both true) on the #$baseSet S. This is because both antisymmetry and asymmetry follow from transitivity and irreflexivity.
> guid: be73e9fb-9c29-11b1-9dad-c379636f7270
> direct instance of: #$ObjectType
> direct specialization of: #$MathematicalOrdering
> direct generalization of: #$TreeOrdering-Strict

#$TreeOrdering tree ordering

> A specialization of #$PartialOrdering. An instance ORDER of #$PartialOrdering is also an instance of #$TreeOrdering just in case the ordering relation R of ORDER orders elements of the #$baseSet S of ORDER into a tree-like structure, so that each pair of elements of S has a common 'R-lower-bound' in S (i.e., for each X, Y in S, there is a Z in S such that R(Z X) and R(Z Y)), and the set of 'R-lower-bounds' of each X in S is ordered in a chain by R (i.e., {Y: Y is in S and R(Y X)} is a chain). Note that an instance of #$TreeOrdering can itself be a chain, i.e., an instance of #$TotalOrdering.
> guid: beb293c3-9c29-11b1-9dad-c379636f7270
> direct instance of: #$ObjectType
> direct specialization of: #$PartialOrdering
> direct generalization of: #$TotalOrdering

#$TreeOrdering-Strict tree ordering strict **COMMENT NOT REVIEWED** **GAFs NOT REVIEWED**

> The collection of all those #$MathematicalOrderings ORDER in which the ordering relation R is irreflexive and transitive on the #$baseSet S of ORDER, and in which every pair of different elements of S has a common 'R-lower-bound', and the set of 'R-lower-bounds' of each X in S is ordered in a (possibly empty) chain by R (i.e., {Y: Y is in S and R(Y X)} is a (possibly empty) chain). Note that the only difference between a #$TreeOrdering and a #$TreeOrdering-Strict is that the ordering relation of the former is reflexive while that of the latter is irreflexive.
> guid: c136761b-9c29-11b1-9dad-c379636f7270
> direct instance of: #$ObjectType
> direct specialization of: #$PartialOrdering-Strict
> direct generalization of: #$TotalOrdering-Strict

#$TotalOrdering total ordering **COMMENT NOT REVIEWED** **GAFs NOT REVIEWED**

> The collection of all those #$PartialOrdering ORDER in which the ordering relation R orders elements of the #$baseSet S of ORDER into a single line. A #$TotalOrdering is sometimes called a 'linear ordering' or a 'chain'. A #$PartialOrdering ORDER is a #$TotalOrdering if all elements of the #$baseSet S of ORDER are comparable by the ordering relation R of ORDER, i.e., for any X and Y in S, either R(X Y) or R(Y X). For example, if you take a set of real numbers and the usual 'greater than or equal to' relation among these numbers, you have a #$TotalOrdering. Note that if you want a 'strict line', i.e., if you want the ordering relation in a total ordering to be irreflexive, see #$TotalOrdering-Strict.
> guid: bf4734b2-9c29-11b1-9dad-c379636f7270
> direct instance of: #$ObjectType
> direct specialization of: #$DistributiveLattice #$TreeOrdering

#$TotalOrdering-Strict total ordering - strict

> A specialization of #$TreeOrdering-Strict. Each instance of #$TotalOrdering-Strict is an ordering in which the ordering relation R orders elements of the #$baseSet S into a strict line. An instance ORDER of #$PartialOrdering-Strict is an instance of #$TotalOrdering-Strict just in case all elements of the #$baseSet S of ORDER are comparable by the ordering relation R of ORDER; i.e., if X and Y are elements of S, then either R(X Y) or X = Y or R(Y X). One example of a #$TotalOrdering-Strict is the set of real numbers with the usual 'smaller than' relation on those numbers.
> guid: be4967b0-9c29-11b1-9dad-c379636f7270
> direct instance of: #$ObjectType
> direct specialization of: #$TreeOrdering-Strict
> direct generalization of: #$WellOrdering

#$WellOrdering well-orderings **COMMENT NOT REVIEWED** **GAFs NOT REVIEWED**

> The collection of all those linear #$MathematicalOrderings that are called 'well-orderings' because they (and their subsets) have 'first' or 'smallest' members. Let ORDER be a #$TotalOrdering-Strict, and let S be the #$baseSet of ORDER and R the ordering relation. ORDER is a #$WellOrdering if every nonempty subset SUBSET of S has its 'R-smallest' member, i.e., there is an X in SUBSET such that R(X Y) for all Y in SUBSET except X. For example, if you take the set of all natural numbers and the usual less-than relation among these numbers then you have a #$WellOrdering because there is a lowest natural number, while if you take the set of all integers (including negative ones) and the usual less-than relation among these numbers then you have a #$TotalOrdering that is not a #$WellOrdering because there is no lowest negative number.
> guid: bfa6adcd-9c29-11b1-9dad-c379636f7270
> direct instance of: #$ObjectType
> direct specialization of: #$TotalOrdering-Strict
> direct generalization of: #$ListWithoutRepetition

#$ListWithoutRepetition lists without repetition **COMMENT NOT REVIEWED** **GAFs NOT REVIEWED**

> The collection of all those #$Lists, for each of which, no element appears more than once in the list. A #$ListWithoutRepetition is sometimes called an 'OSET'. Note that a #$ListWithoutRepetition amounts to a finite #$TotalOrdering-Strict. (In general a #$List and #$Series may have repeated elements.) See also #$SeriesWithoutRepetition.
> guid: bdb9081e-9c29-11b1-9dad-c379636f7270
> direct instance of: #$ObjectType
> direct specialization of: #$WellOrdering #$List

#$Multigraph multigraph **COMMENT NOT REVIEWED** **GAFs NOT REVIEWED**

> An instance of #$PathSystemType-Structural and a subcollection of #$PathSystem. Each instance of #$Multigraph is an instance of #$PathSystem in which the only points are nodes in the system and all paths are made of links (i.e., no intermediate points along links). Sometime such a system is called a graph or multi-graph in graph theory. A #$Multigraph consists of nodes interconnected by links, with loops on single nodes allowed, and with multiple links between the same two nodes also allowed. (For a graph with no parallel links and no loops, see #$SimpleGraph-GraphTheoretic.
> guid: bde212ef-9c29-11b1-9dad-c379636f7270
> direct instance of: #$PathSystemType-Structural
> direct specialization of: #$PointFinitePathSystem #$RelationalStructure
> direct generalization of: #$DirectedMultigraph #$SimpleGraph-GraphTheoretic

#$SimpleGraph-GraphTheoretic simple graphs **COMMENT NOT REVIEWED** **GAFs NOT REVIEWED**

> The collection of all #$PathSystems that are instances of both #$SimplePathSystem and #$Multigraph. Each instance of #$SimpleGraph-GraphTheoretic is a 'graph', as studied in graph theory, in which there are neither loops nor multiple links between the same pair of nodes.
> guid: be269b3d-9c29-11b1-9dad-c379636f7270
> direct instance of: #$PathSystemType-Structural
> direct specialization of: #$SimplePathSystem #$Multigraph
> direct generalization of: #$DirectedGraph

#$DirectedMultigraph directed multigraph

> A specialization of both #$DirectedPathSystem and #$Multigraph. Each instance of #$DirectedMultigraph is a multigraph in which every link has one direction. Note that there can be loops and multiple links between a pair of nodes in a given instance of #$DirectedMultigraph.
> guid: c0ba0c32-9c29-11b1-9dad-c379636f7270
> direct instance of: #$PathSystemType-Structural
> direct specialization of: #$DirectedPathSystem #$Multigraph
> direct generalization of: #$DirectedGraph

#$DirectedGraph directed graph **COMMENT NOT REVIEWED** **GAFs NOT REVIEWED**

> The collection of all directed simple graphs, i.e., node-and-link structure in which every link has one direction and no multiple links (between a pair of nodes) or loops are allowed, as studied in graph theory. This is the intersection of #$SimpleGraph-GraphTheoretic and #$DirectedMultigraph, which is the same as the intersection of #$SimpleGraph-GraphTheoretic and #$DirectedPathSystem.
> guid: beb3df26-9c29-11b1-9dad-c379636f7270
> direct instance of: #$PathSystemType-Structural
> direct specialization of: #$DirectedMultigraph #$SimpleGraph-GraphTheoretic
> direct generalization of: #$DirectedAcyclicGraph

#$DirectedAcyclicGraph Directed Acyclic Graph **COMMENT NOT REVIEWED** **GAFs NOT REVIEWED**

> The collection of all those #$DirectedGraphs (node-and-link structures in which each link has one direction) each of which has no directed cycle in it. This is the intersection of #$DirectedGraph and #$DirectedAcyclicPathSystem (which is the same as the intersection of #$SimpleGraph-GraphTheoretic and #$DirectedAcyclicPathSystem). A #$DirectedAcyclicGraph is often used as a representation of a #$PartialOrdering.
> guid: bed5ca59-9c29-11b1-9dad-c379636f7270
> direct instance of: #$PathSystemType-Structural
> direct specialization of: #$DirectedGraph #$DirectedAcyclicPathSystem

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