Two definability results in the equational context

Abstract

Abstract.Let r be a type bounded by an infinite regular cardinal a , V be a variety in z , x' Ç t and V the class of all r'-reducts of the algebras in V . We show that the operations in t\t' are explicitely definable in V by pure formulas (i.e. existential-positive without disjunction) if and only if they are implicitely definable and V is closed under unions of o-chains (if and only if every t'-homomorphisms between algebras in V are r-homomorphisms, as J. Isbell has shown). It follows that the operations in t\t' are equivalent (in V) to r'-terms if and only if every algebra in the (t'-) variety generated by V has a unique r-expansion in V .As usual, ordinals will be identified with the set of smaller ordinals, and cardinals with initial ordinals. Definitions 1. (i)A type x is a set of operation symbols, each with an assigned arity (some cardinal), t is bounded by the infinite regular cardinal a if the arities of its operation symbols are all < a.(ii) If C is a class of (t-) algebras, an a-sequence (in C) is an a-chain {21 c 21,, |0 < p. < y < a} (where " ç " means "subalgebra") of algebras in C each of which is a section of a fixed 21 e C (i.e. 21 ç 21 for every p. and these inclusions have left inverse homomorphisms).(iii) If x ç t and 21 is a t-algebra, we will denote, if the context is clear, by 2l' (instead of the usual 21 |~T,) the r'-reduct [2] of 21. Similarly, if C is a class of T-algebras, C' will be an abbreviation for C["r,= {2l'|2l e C} .