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The equational definability of truth predicates

100%

Raftery J. G.

tom Nr 41,spec. iss.

95-149

Let S be a structural consequence relation, not assumed to be protoalgebraic. It is proved that the following con- ditions on S are equivalent, where `algebra' means algebra in the signature of S: (1) S is truth-equational, i.e., the truth predicate of the class of reduced matrix models of S is explicitly denable by some xed set of unary equations. (2) The Leibniz operator of S is completely order re ecting on all algebras, i.e., for any set of S{lters F [ fGg of an algebra, if T [F] G then TF G. (3) The Leibniz operator is completely order re ecting on the theories of S. (4) The Suszko operator of S is injective on all algebras. It makes no dierence to the meaning of (1) whether `reduced' is interpreted as Leibniz-reduced or as Suszko-reduced. For the class of Suszko-reduced matrix models of S, (4) ) (1) says that the im- plicit denability of the truth predicate entails its equational de- nability. Previously, this was known only for protoalgebraic sys- tems. The corresponding assertion for the Leibniz-reduced models is shown to be false, i.e., global injectivity of the Leibniz operator does not entail truth-equationality.

On the variety generated by involutive pocrims

100%

Raftery J. G.

Reports on Mathematical Logic

2007

tom No. 42

71-86

An involutive pocrim (a.k.a. an L0-algebra) is a residuated integral partially ordered commutative monoid with an involution operator, considered as an algebra. It is proved that the variety generated by all involutive pocrims satises no nontrivial idempotent Maltsev condition. That is, no nontrivial (logical and, logical or, o) -equation holds in the congruence lattices of all involutive pocrims. This strengthens a theorem of A.Wroński. The result survives if we restrict the generating class to totally ordered involutive pocrims.

Irreducible residuated semilattices and finitely based varieties

51%

Galatos N. , Olson J. S. , Raftery J. G.

tom No. 43

805-108

This paper deals with axiomatization problems for varieties of residuated meet semilattice-ordered monoids (RSs). An internal characterization of the finitely subdirectly irreducible RSs is proved, and it is used to investigate the varieties of RSs within which the finitely based subvarieties are closed under fi- nite joins. It is shown that a variety has this closure property if its finitely subdirectly irreducible members form an elementary class. A syntactic characterization of this hypothesis is proved, and examples are discussed.