Two definability results in the equational context

Authors:
M. Hébert, R. N. McKenzie and G. E. Weaver

Journal:
Proc. Amer. Math. Soc. **107** (1989), 47-53

MSC:
Primary 08B05; Secondary 03C05, 03C40

DOI:
https://doi.org/10.1090/S0002-9939-1989-0975648-1

MathSciNet review:
975648

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Abstract: Let be a type bounded by an infinite regular cardinal , be a variety in and the class of all -reducts of the algebras in . We show that the operations in are explicitely definable in by *pure formulas* (i.e. existential-positive without disjunction) if and only if they are implicitely definable and is closed under unions of -chains (if and only if every -homomorphisms between algebras in are -homomorphisms, as J. Isbell has shown). It follows that the operations in are equivalent (in ) to -terms if and only if every algebra in the variety generated by has a unique -expansion in .

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DOI:
https://doi.org/10.1090/S0002-9939-1989-0975648-1

Article copyright:
© Copyright 1989
American Mathematical Society