Lattice Of Interpretability Types Research Articles

On the primeness of locally finite idempotent 3-permutability

In this paper, we show that for any finite idempotent non-congruence 3-permutable algebra \(\mathbf {A}\), it is always the case that \({{\,\mathrm{HSP}\,}}(\mathbf {A})\) contains an algebra carrying a binary reflexive compatible relation having a certain shape, which we have called a 2-dimensional special Hagemann relation with middle part. As a result of this property, we are able to show that the join of any pair of locally finite idempotent non-congruence 3-permutable varieties in the lattice of interpretability types, fails to be congruence 3-permutable, yielding a primeness argument for this specific setting.

Taylor’s modularity conjecture holds for linear idempotent varieties

The “Modularity Conjecture” is the assertion that the join of two nonmodular varieties in the lattice of interpretability types is nonmodular. We establish the veracity of this conjecture for the case of linear idempotent varieties. We also establish analogous results concerning n-permutability for some n, and the satisfaction of nontrivial congruence identities.

The Lattice of Interpretability Types of Cantor Varieties

For integers 1 ≤ m < n, a Cantor variety with m basic n-ary operations ωi and n basic m-ary operations λk is a variety of algebras defined by identities λk(ω1(\(x\)), ... , ωm(\(\bar x\))) = \(x\)k and ωi(λ1(\(\bar y\)), ... ,λn(\(\bar y\))) = yi, where \(\bar x\) = (x1., ... , xn) and \(\bar y\) = (y1, ... , ym). We prove that interpretability types of Cantor varieties form a distributive lattice, ℂ, which is dual to the direct product ℤ1 × ℤ2 of a lattice, ℤ1, of positive integers respecting the natural linear ordering and a lattice, ℤ2, of positive integers with divisibility. The lattice ℂ is an upper subsemilattice of the lattice \(\mathbb{L}^{\operatorname{int} } \) of all interpretability types of varieties of algebras.

Some universal properties of the category of clones

In [6], O. C. Garcia and W. Taylor asked if the breadth of the lattice of interpretability types of varieties is uncountable. The present paper solves the problem by two different constructions. Both of them show that any cardinal number is the cardinality of an antichain in the named lattice and that the existence of a proper class antichain is equivalent to the negation of Vopěnka's principle. The first construction gives in a way a minimal solution of the problem, whereas the second one gives stronger results about the category of clones.

Interpretations into monadic algebras

In [3], O. C. Garcia and W. Taylor make an in depth study of the lattice of interpretability types of varieties first introduced by W. Neumann [5]. In this lattice several varieties are identified so in order to distinguish them and understand the fine structure of the lattice, we propose the study of the interpretations between them, in particular, how many there are and what these are. We prove, among other things, that there are eight interpretations from the variety of Monadic algebras into itself.

Some Prime Elements in the Lattice of Interpretability Types

Some Prime Elements in the Lattice of Interpretability Types

Some prime elements in the lattice of interpretability types

A general theorem is proved which implies that the types of PA \text {PA} (Peano Arithmetic), ZF \text {ZF} (Zermelo-Fraenkel Set Theory) and GB \text {GB} (Gödel-Bernays Set Theory) are prime in the lattice of interpretability types.

A lattice of interpretability types of theories

We consider first-order logic only. A theory S will be called locally interpretable in a theory T if every theorem of S is interpretable in T. If S is locally interpretable in T and T is consistent then S is consistent. Most known relative consistency proofs can be viewed as local interpretations. The classic examples are the cartesian interpretation of the elementary theorems of Euclidean n-dimensional geometry into the first-order theory of real closed fields, the interpretation of the arithmetic of integers (rational numbers) into the arithmetic of positive integers, the interpretation of ZF + (V = L) into ZF, the interpretation of analysis into ZFC, relative consistency proofs by forcing, etc. Those interpretations are global. Under fairly general conditions local interpretability implies global interpretability; see Remarks (7), (8), and (9) below.We define the type (interpretability type) of a theory S to be the class of all theories T such that S is locally interpretable in T and vice versa. There happen to be such types and they are partially ordered by the relation of local interpretability. This partial ordering is of lattice type and has the following form:The lattice is distributive and complete and satisfies the infinite distributivity law of Brouwerian lattices:We do not know if the dual lawis true. We will show that the lattice is algebraic and that its compact elements form a sublattice and are precisely the types of finitely axiomatizable theories, and several other facts.