Recently we studied the topic about structures at school, but I didn't quite get it. So my task is to check if a given set is an universe for a substructure of F (the structure of the natural numbers), where F = (N,+,*), N are the natural numbers and + and . are the standard addition and multiplication. Let's say the given set I should check for is {0,1} what steps should I take to determine if this set is a universe for a substructure of F.
How to define if a given set is an universe for a substructure?
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$\begingroup$
group-theory
group-homomorphism
1 Answers
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If $U$ is the universe of a structure for a language $L$, and $V \subseteq U$, then for $V$ to be the universe for a substructure, you must verify that $V$ is closed under all functions in the language $L$ (including any constants, which are considered 0-ary functions). If $V$ wasn't closed under some function, then that function from $L$ wouldn't be defined in $V$, so $V$ wouldn't be a substructure. In your case, the functions are + and *. Relations are automatically inherited from the superstructure (since their outputs are true/false rather than an element of the universe), and do not need to be checked.