order_theory.theory.txt

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ORDER THEORY ==>                  #Math branch about order

PARTIAL ORDER ==>                 #Binary relation that is:
                                  #  - transitive
                                  #  - antisymmetric
                                  #  - homogeneous

STRICT PARTIAL ORDER ==>          #Partial order that is not reflexive.
                                  #Also named "irreflexive", "strong".
                                  #E.g. <

NON-STRICT PARTIAL ORDER ==>      #Partial order that is reflexive.
                                  #Also named "reflexive", "weak".
                                  #E.g. <=

TOTAL ORDER ==>                   #Non-strict partial order where every element is strongly connected ("total")

GREATEST ELEMENT ==>              #Also named "top" or "unit"
                                  #Noted ⊤  or 1
                                  #Value of a given set >= any others
LEAST ELEMENT ==>                 #Also named "bottom" or "zero"
                                  #Noted ⊥  or 0
                                  #Inverse of greatest element

MAXIMAL ELEMENT ==>               #Value of a given set not < any others
                                  #If set is:
                                  #  - totally ordered: same as greatest element
                                  #  - partially ordered: there is 0-1 greatest element, but can be 0-n maximal elements
MINIMAL ELEMENT ==>               #Inverse of maximal element

MAXIMUM|MINIMUM ==>               #Maximal|minimal element of a totally ordered set, i.e. also greatest|least element
EXTREMUM ==>                      #Maximum or minimum

ABSOLUTE MAXIMUM|MINIMUM ==>      #Also named "global maximum|minimum"
                                  #Maximal|minimal element of a function's codomain
RELATIVE MAXIMUM|MINIMUM ==>      #Also named "local maximum|minimum"
                                  #Like abolute maximum but only for a subset of the function's domain+codomain

TIGHT UPPER BOUND ==>             #Also named "supremum", "join" or "least|sharp|optimal upper bound"
                                  #Lowest value >= any value of a given set
                                  #Not always same as greatest|maximal element
                                  #  - e.g. infinite sets have no greatest|maximal element, but can have tight upper bound
                                  #But:
                                  #  - greatest element is always tight upper bound
                                  #  - one maximal element is always tight upper bound
TIGHT LOWER BOUND ==>             #Also named "infimum", "meet" or "greatest|sharp|optimal lower bound"
                                  #Inverse of tight upper bound.

UPPER BOUND ==>                   #Also named "majorant"
                                  #Any value >= any value of a given set
                                  #That value is said to "majorize"/"bound from above" the set.
LOWER BOUND ==>                   #Also named "minorant"
                                  #Inverse of upper bound
                                  #That value is said to "minorize"/"bound from below" the set.

FUNCTION UPPER|LOWER BOUND ==>    #Upper|lower bound of a function's codomain.

UPPER|LOWER BOUNDING FUNCTION ==> #Function f where any f(x) >= g(x), i.e. f is upper bound of g
                                  #Or inverse for lower bound