Partially specified elements of a group?

  • Context: Graduate 
  • Thread starter Thread starter Stephen Tashi
  • Start date Start date
  • Tags Tags
    Elements Group
Click For Summary
SUMMARY

The discussion centers on the concept of "partially specified elements" within group theory, specifically regarding their representation as permutations on a set of group elements. Participants define a partial specification as a set of ordered pairs that incompletely defines a one-to-one function, exemplified by the pairs {(a,b),(c,a)} for the group elements {a,b,c,d}. The discussion clarifies that while these partial specifications can be considered relations, they are more specific than general relations, which do not necessarily represent a subset of ordered pairs of a function.

PREREQUISITES
  • Understanding of group theory concepts
  • Familiarity with permutations and functions
  • Knowledge of mathematical relations and their properties
  • Basic grasp of ordered pairs and their significance in mathematics
NEXT STEPS
  • Research the properties of permutations in group theory
  • Study the concept of relations in mathematics
  • Explore the implications of partial functions in algebra
  • Learn about the composition of mappings in group theory
USEFUL FOR

Mathematicians, students of abstract algebra, and anyone interested in advanced group theory concepts and their applications in mathematical functions.

Stephen Tashi
Science Advisor
Homework Helper
Education Advisor
Messages
7,864
Reaction score
1,605
Is there a technical term in group theory for (what I would call) partially specified elements of a group?

I mean "partially specified" in the following sense:

An elements of a group acts as permuation on the set of elements of the group. So a group element can be considered to be a function that is specified by a set of ordered pairs. Suppose we have a set of ordered pairs of group elements that incompletely specifies a 1-1 function on the group. For example if the group elements are {a,b,c,d} we might have the set of ordered pairs { (a,b),(c,a)} which is missing a specification for (b,?) and (d,?).

If A is a partial specification then reversing the ordered pairs in A give a different partial specification which one might call the inverse specification. If A and B are two partial specification then the product can be defined as the partial specification given by forming the composition of the two mappings, insofar as we can do so from the given ordered pairs in A and B.
 
Physics news on Phys.org
As I know it, it's called a "relation".

From wikipedia: "Formally, a relation is a set of n-tuples of equal degree."
 
I like Serena said:
As I know it, it's called a "relation".

I agree that it is a relation, but this type of partial specification is more specific than a general relation. A general relation need not even be a subset of the ordered pairs of a function.
 

Similar threads

Graduate Probability distributions in an ordered set of extracted elements
  • · Replies 2 ·
Replies
2
Views
2K
Insights Groups, The Path from a Simple Concept to Mysterious Results
  • · Replies 17 ·
Replies
17
Views
10K
Undergrad Does the definition of cardinality assume distinguishability?
  • · Replies 9 ·
Replies
9
Views
2K
Undergrad Add an exponential number of elements, what will be the final cardinality?
  • · Replies 16 ·
Replies
16
Views
2K
MHB Partial Order Relation on a Functions Set
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Direct limit of multiverse models of ZFC
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Finding the Number of Sets in Two Groups: A Simplified Problem
  • · Replies 2 ·
Replies
2
Views
1K
Undergrad Question about "A Group Epimorphism is Surjective"
  • · Replies 13 ·
Replies
13
Views
2K
Undergrad Can an ordered pair have identical elements?
  • · Replies 14 ·
Replies
14
Views
4K
Undergrad Observational Studies: What to Believe
  • · Replies 3 ·
Replies
3
Views
3K