Definition
Given two structures A and B of the same signature σ, A is said to be a weak substructure of B, or a weak subalgebra of B, if
- the domain of A is a subset of the domain of B,
- f A = f B | An for every n-ary function symbol f in σ, and
- R A R B An for every n-ary relation symbol R in σ.
A is said to be a substructure of B, or a subalgebra of B, if A is a weak subalgebra of B and, moreover,
- R A = R B An for every n-ary relation symbol R in σ.
If A is a substructure of B, then B is called a superstructure of A or, especially if A is an induced substructure, an extension of A.
Read more about this topic: Substructure
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