On the fullness of certain functors

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} .

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“…This leads to results analogous to some of the ones in II and III for situations outside model theory ( [6]). …”

Section: Definitions 2 (I)supporting

confidence: 77%

Two definability results in the equational context

Hébert¹,

McKenzie²,

Weaver³

1989

Proc. Amer. Math. Soc.

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“…Note finally that there cannot be such a weak correspondence between limit-invariant theories T and universal strict Horn theories T*: indeed, Corollary 1 of [9] implies that with such theories, a forgetful functor M_ (T_*)~M_ (T) which is bijective on objects is necessarily an isomorphism, and this has been seen to be impossible above. For example in the case of Theorem 7, take At = {VX(fl(X)~VN~,(X))IN is finite, {fl, ~, ] n ~ N} c_ A At}; to obtain T* in A~.…”

Section: ~ {(Pi(k)(c) V Pick)(c)) ^ (P J(k)(c) V P J(k)(c))mentioning

confidence: 99%

Syntactic characterization of closure under connected limits

Hébert

1991

Arch Math Logic

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We give a syntactic characterization of (finitary) theories whose categories of models are closed under the formation of connected limits (respectively the formation of pullbacks and substructures) in the category of all structures. They are also those theories whose consistent extensions by new atomic facts admit in each component an initial structure (respectively an initial term structure), and also those T for which _M(_T) is locally finitely multi-presentable in a canonical way. We also show that these two properties of theories are nonuniform.* 1980

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Hébert

1998

Applied Categorical Structures

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On the fullness of certain functors

Cited by 3 publications

References 17 publications

Two definability results in the equational context

Two definability results in the equational context

Syntactic characterization of closure under connected limits

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