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Deductive systems of BCK-algebras

Sergio A. Celani (2004)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

In this paper we shall give some results on irreducible deductive systems in BCK-algebras and we shall prove that the set of all deductive systems of a BCK-algebra is a Heyting algebra. As a consequence of this result we shall show that the annihilator F * of a deductive system F is the the pseudocomplement of F . These results are more general than that the similar results given by M. Kondo in [7].

Definability of small puncture sets

Andrés Eduardo Caicedo, John Daniel Clemens, Clinton Taylor Conley, Benjamin David Miller (2011)

Fundamenta Mathematicae

We characterize the class of definable families of countable sets for which there is a single countable definable set intersecting every element of the family.

Definability within structures related to Pascal’s triangle modulo an integer

Alexis Bès, Ivan Korec (1998)

Fundamenta Mathematicae

Let Sq denote the set of squares, and let S Q n be the squaring function restricted to powers of n; let ⊥ denote the coprimeness relation. Let B n ( x , y ) = ( x + y x ) M O D n . For every integer n ≥ 2 addition and multiplication are definable in the structures ⟨ℕ; Bn,⊥⟩ and ⟨ℕ; Bn,Sq⟩; thus their elementary theories are undecidable. On the other hand, for every prime p the elementary theory of ⟨ℕ; Bp,SQp⟩ is decidable.

Definable Davies' theorem

Asger Törnquist, William Weiss (2009)

Fundamenta Mathematicae

We prove the following descriptive set-theoretic analogue of a theorem of R. O. Davies: Every Σ¹₂ function f:ℝ × ℝ → ℝ can be represented as a sum of rectangular Σ¹₂ functions if and only if all reals are constructible.

Definable hereditary families in the projective hierarchy

R. Barua, V. Srivatsa (1992)

Fundamenta Mathematicae

We show that if ℱ is a hereditary family of subsets of ω ω satisfying certain definable conditions, then the Δ 1 1 reals are precisely the reals α such that β : α Δ 1 1 ( β ) . This generalizes the results for measure and category. Appropriate generalization to the higher levels of the projective hierarchy is obtained under Projective Determinacy. Application of this result to the Q 2 n + 1 -encodable reals is also shown.

Definable orthogonality classes in accessible categories are small

Joan Bagaria, Carles Casacuberta, A. R. D. Mathias, Jiří Rosický (2015)

Journal of the European Mathematical Society

We lower substantially the strength of the assumptions needed for the validity of certain results in category theory and homotopy theory which were known to follow from Vopěnka’s principle. We prove that the necessary large-cardinal hypotheses depend on the complexity of the formulas defining the given classes, in the sense of the Lévy hierarchy. For example, the statement that, for a class 𝒮 of morphisms in a locally presentable category 𝒞 of structures, the orthogonal class of objects is a small-orthogonality...

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