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Encyclopedia > Elementary substructure

In model theory, given two structures M and N in the same language L, we say that M is an elementary substructure of N (notated sometimes M < N) if


1. M is a substructure of N, and


2. for every finite tuple , for every formula of the language L, we have that if and only if .


The second part may also be presented as saying that

ThL(M)(M) = ThL(M)(N).

The Tarski-Vaught test is very useful in determining whether, given a pair , M is an elementary substructure of N.


  Results from FactBites:
 
Elementary - Wikipedia, the free encyclopedia (340 words)
The elementary charge (symbol e or sometimes q) is the electric charge carried by a single proton, or equivalently, the negative of the electric charge carried by a single electron.
Elementary algebra is the most basic form of algebra taught to students who are presumed to have no knowledge of mathematics beyond the basic principles of arithmetic.
Elementary matrix transformations or elementary row and column transformations are linear transformations which are normally used in Gaussian elimination to solve a set of linear equations.
elementary particle: Definition and Much More from Answers.com (0 words)
The study of elementary particles is one of the frontiers of modern physics and is associated with the building of particle accelerators.
If an elementary particle truly has no substructure, then it is one of the basic particles of the universe from which all larger particles are made.
A central feature in elementary particle theory is the early 20th century idea of "quanta", which revolutionised the understanding of electromagnetic radiation and brought about quantum mechanics.
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