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2020 Mathematics Subject Classification: Primary: 68P05 [MSN][ZBL]

An interpretation only defined on the ground terms t\in T(\Sigma) of a signature \Sigma is called an evaluation. Since interpretations are \Sigma-algebra-morphisms, evaluations are \Sigma-algebra-morphisms as well. Furthermore, evaluations are uniquely determined, i.e. there exists exactly one mapping e\colon T(\Sigma)\longrightarrow A. This specific property has remarkable consequences. Consider for example a \Sigma-algebra-morphism f\colon A\longrightarrow B between \Sigma-algebras A and B. Then the equality e_A=f\circ e_B holds for evaluations e_A and e_B. In effect, each assignement can be extended to a functor between the term algebra T(\Sigma) and A.

For reasons of simplicity, the application of the (uniquely determined) evaluation e\colon T(\Sigma)\longrightarrow A to a term t\in T(\Sigma) is often designated as t^A := e(t).

References

[EM85] H. Ehrig, B. Mahr: "Fundamentals of Algebraic Specifications", Volume 1, Springer 1985
How to Cite This Entry:
Evaluation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Evaluation&oldid=29687