✳flâneur — a map of the web's best reading

Elementary equivalence - Wikipedia

en.wikipedia.org · 1,705 words · saved by 1 readers

In model theory, a branch of mathematical logic, two structures M and N of the same signature σ are called elementarily equivalent if they satisfy the same first-order σ-sentences. If N is a substructure of M, one often needs a stronger condition. In this case N is called an elementary substructure of M if every first-order σ-formula φ(a1, …, an) with parameters a1, …, an from N is true in N if and only if it is true in M. If N is an elementary substructure of M, then M is called an elementary extension of N. An embedding h: N → M is called an elementary embedding of N into M if h(N) is an elementary substructure of M. A substructure N of M is elementary if and only if it passes the Tarski–Vaught test: every first-order formula φ(x, b1, …, bn) with parameters in N that has a solution in M also has a solution in N when evaluated in M. One can prove that two structures are elementarily equivalent with the Ehrenfeucht–Fraïssé games. Elementary embeddings are used in the study of large car

Elementary equivalence - Wikipedia Jump to content From Wikipedia, the free encyclopedia Concept in model theory This article includes a list of references , related reading , or external links , but its sources remain unclear because it lacks inline citations . Please help improve this article by introducing more precise citations. ( February 2023 ) ( Learn how and when to remove this message ) In model theory , a branch of mathematical logic , two structures M and N of the same signature σ are called elementarily equivalent if they satisfy the same first-order σ -sentences . If N is a substr

Explore this link on the map →