Discriminator logics (Research announcement)

A discriminator logic is the 1-assertional logic of a discriminator variety V having two constant terms 0 and 1 such that V ⊨ 0 1 iff every member of V is trivial. Examples of such logics abound in the literature. The main result of this research announcement asserts that a certain non-Fregean deductive system SBPC, which closely resembles the classical propositional calculus, is canonical for the class of discriminator logics in the sense that any discriminator logic S can be presented (up to definitional equivalence) as an axiomatic extension of SBPC by a set of extensional logical connectives taken from the language of S. The results outlined in this research announcement are extended to several generalisations of the class of discriminator logics in the main work.

Quasi-discriminator varieties

We generalize the notion of discriminator variety in such a way as to capture several varieties of algebras arising mainly from fuzzy logic. After investigating the extent to which this more general concept retains the basic properties of discriminator varieties, we give both an equational and a purely algebraic characterization of quasi-discriminator varieties. Finally, we completely describe the lattice of subvarieties of the pure pointed quasi-discriminator variety, providing an explicit equational base for each of its members.

Decompositions of dual discriminator varieties

A necessary and sufficient condition for a dual discriminator variety to be the product of a discriminator variety and a dual discriminator variety is given, along with an example of such a variety.

A structure theorem for strongly abelian varieties with few models

By a variety, we mean a class of structures in some language containing only function symbols which is equationally defined or equivalently is closed under homomorphisms, submodels and products.If K is a class of -structures then I(K, λ) denotes the number of nonisomorphic models in K of cardinality λ. When we say that K has few models, we mean that I(K,λ) < 2λ for some λ > ∣∣. If I(K,λ) = 2λ for all λ > ∣∣, then we say K has many models. In [9] and [10], Shelah has shown that for an elementary class K, having few models is a strong structural condition.Before we give the definition of strongly abelian, let us motivate how it arises in this context. A variety is locally finite if every finitely generated algebra in is finite. If and are subvarieties of then = ⊗ means that is the variety generated by and and moreover there is a term τ(x, y) so that τ(x, y) = x holds in and τ(x, y) = y holds in . is called the varietal product of and . As a consequence, if = ⊗ then for every M ∈ there is a unique (up to isomorphism) A ∈ and B ∈ so that M ≅ A × B.In [4], McKenzie and Valeriote proved the following theorem.Theorem 0.1. Ifis a locally finite decidable variety, then there are three subvarieties of, , and, so that = ⊗ ⊗ andis an affine variety, is a strongly abelian variety andis a discriminator variety.For the exact definitions of the terms affine and discriminator one can see [4]; however, for us here it is important to know that an affine variety is polynomially equivalent to a variety of left R-modules over some ring R and that any nontrivial discriminator variety contains an algebra whose complete theory is unstable.

Finitely generic models of TUH, for certain model companionable theories T

The starting point of this work was Saracino and Wood's description of the finitely generic abelian ordered groups [S-W].We generalize the result of Saracino and Wood to a class ∑UH of subdirect products of substructures of elements of a class ∑, which has some relationships with the discriminator variety V(∑t) generated by ∑. More precisely, let ∑ be an elementary class of L-algebras with theory T. Burris and Werner have shown that if ∑ has a model companion then the existentially closed models in the discriminator variety V(∑t) form an elementary class which they have axiomatized. In general it is not the case that the existentially closed elements of ∑UH form an elementary class. For instance, take for ∑ the class ∑0 of linearly ordered abelian groups (see [G-P]).We determine the finitely generic elements of ∑UH via the three following conditions on T:(1) There is an open L-formula which says in any element of ∑UH that the complement of equalizers do not overlap.(2) There is an existentially closed element of ∑UH which is an L-reduct of an element of V(∑t) and whose L-extensions respect the relationships between the complements of the equalizers.(3) For any models A, B of T, there exists a model C of TUH such that A and B embed in C.(Condition (3) is weaker then “T has the joint embedding property”. It is satisfied for example if every model of T has a one-element substructure. Condition (3) implies that ∑UH has the joint embedding property and therefore that the class of finitely generic elements of ∑UH is complete.)