Wikipedia defines a group as "an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element." I keep thinking that there is a connection to this definition and a relation on a set, but I'm not sure what it is. Obviously, relations and operators are connected. Can groups be defined in terms of sets and relations? I am new to this, and the Wikipedia article is over my head.
Can a group be defined in terms of a relation on a set?
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group-theory
elementary-set-theory
relations
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0Well, everything is connected. I'm not really sure what you're looking for. – 2011-09-17
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0@Qiaochu Yuan In other words, is it possible to define a group as "an algebraic structure consisting of a set together with a relation..."? – 2011-09-17
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1This seems to be relevant (see the answer): http://math.stackexchange.com/questions/4009/confusion-between-operation-and-relation-clarification-needed. – 2011-09-17
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0@Srivatsan Narayanan Yes, that is relevant, thank you. So it would seem that the answer to my question is _no, relations are more general than operators_, and the question becomes, is there there a name for "an algebraic structure consisting of a set together with a relation..."? – 2011-09-17
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2You should look into a branch of mathematical logic called model theory. Within some logical language (often work in first order logic), a set (called a domain) together with relations, and special relations called constants and functions form something called a structure. – 2011-09-17
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0@William Chan Thanks, will do. That's part of what I was looking for--what's the next step of abstraction above groups. – 2011-09-17
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0For a reference on groups given in terms of generators and relations, see the book by Manus, Karrass and Solitar entitled "Combinatorial group theory: presentations of groups in terms of generators and relations". – 2011-09-17
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A group can be defined as a set and a relations.
Note that a relation $R$ on a set $G$ is any subset of $G \times G$. A function is a relation on $G \times G$ such that if $(a,b) \in f$ and $(a,c) \in f$, then $b = c$.
Therefore a group is a set $G$ with a relation $*$ which happens to be a function. Moreover, this function satisfies some properties like associativity, etc. Also as is typical in model theory, you often say a group is a set, with a binary function $*$, and a constant $e$, which represents the identity. Again, the constant can still be thought of as a unary relation. You can also define a group to include a symbol for taking an inverse. This can still be thought of as a relation since it is a unary function.
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0Thanks; an extremely helpful answer. – 2011-09-17
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3Be careful: a binary relation on $G \times G$ would be a subset of $G \times G \times G \times G$. A binary operation is a ternary relation, i.e. a subset of $G \times G \times G$. – 2011-09-17