Classes without the amalgamation property

Comer, Stephen D.

doi:10.2140/pjm.1969.28.309

Cited by 25 publications

(21 citation statements)

References 5 publications

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“…The interpolation theorem stated in this way fails for both RRA and RA [4]. One can similarly formulate a Beth definability theorem for RRA and RA (restricted to implicitly defined binary and unary relations only).…”

Section: Interpolation and Definabilitymentioning

confidence: 99%

Relation Algebra with Binders

Marx

2001

Journal of Logic and Computation

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The language of relation algebras is expanded with variables denoting individual elements in the domain and with the ↓ binder from hybrid logic. Every elementary property of binary relations is expressible in the resulting language, something which fails for the relation algebraic language. That the new language is natural for speaking about binary relations is indicated by the fact that both Craig's Interpolation, and Beth's Definability theorems hold for its set of validities. The paper contains a number of worked out examples.

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Section: Interpolation and Definabilitymentioning

confidence: 99%

Relation Algebra with Binders

Marx

2001

Journal of Logic and Computation

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“…Comer's results on CA n and Df n were obtained as corollaries from two much stronger theorems (Theorems 1 and 2 in [3]). The main lemma on which these theorems depend is [3]: Lemma 3. Lemma 5 below is our stronger version of Comer's Lemma.…”

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confidence: 97%

“…Apply a similar argument, but now use a variable d which is interpreted in the algebras A i as the diagonal d 01 Relation Algebras. [3]: Theorem 1 implies that for any class of relation algebras between RA and RRA, the embedding property fails. Using our stronger lemma we can extend the range of relational type algebras for which this holds.…”

mentioning

confidence: 99%

“…REMARK 1. In [3], the similarity type of diagonal free cylindric algebras is called cylindrification type, a full diagonal free cylindric set algebra is called a full cylindrification set algebra, etc. The class Df α is denoted by Cy α , and RDf α by RCy α .…”

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confidence: 99%

“…Using our stronger lemma we can extend the range of relational type algebras for which this holds. We outline the differences in the proof, for the full argument we refer to [3]. In order to obtain an equational definitional embedding into the class K 2 defined in Theorem 3, it is only required that the boolean laws hold and that (the composition operator); is normal, additive, and satisfies 1; (x; 1) ≤ (1; x); 1.…”

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confidence: 99%

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Amalgamation in finite dimensional cylindric algebras

Marx¹

2000

Algebra Universalis

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For every finite n > 1, the embedding property fails in the class of all n-dimensional cylindric type algebras which satisfy the following. Their boolean reducts are boolean algebras and two of the cylindrifications are normal, additive and commute. This result also holds for all subclasses containing the representable n-dimensional cylindric algebras. This considerably strengthens a result of S. Comer on CA n and provides a strong counterexample for interpolation in finite variable fragments of first order logic. We provide a new modern proof, using an argument inspired by modal logic.S. Comer showed that for finite n larger than 1, the embedding property fails in any class of cylindric type algebras of dimension n lying between RCA n and CA n [3]. On the other hand, the proof method employed by I. Németi in [9] can be used to show that for any α, the class NCA α (defined by all CA α -axioms except commutativity of the cylindrifications) and its subclass D α of cylindric-relativised set algebras have the strong amalgamation property. In this last class, commutativity only holds in the restricted sense of c i (c j . So some combination of the NCA axioms together with some commutativity of the cylindrifications can be seen as the cause of the failure of the embedding property. It is a natural question then to ask how much commutativity and how much of the other axioms is really needed to obtain Comer's result. It turns out that for every finite n, we only need to ask that the first two cylindrifications are normal, additive and commute in only one direction. So we find a unique reason for the failure of the embedding property for every n. This considerably strengthens Comer's result. The difference lies not only in the required properties of the cylindrifications (e.g., in CA n , they are complemented closure operators, in our case they are just operations), but in particular that for any finite n, the failure of the embedding property is caused by the behaviour of the same two cylindrifications. We provide a new modern proof, using an argument inspired by modal logic. These algebraic results have -by the correspondence theorems developed in [10]-a direct logical meaning: they provide a strong counterexample for interpolation in first order logic with only finitely many variables.We recall the definitions of the above mentioned classes of cylindric type algebras and their operations from [6]. An algebra A= (A, ∧, ∨, −, 0, 1, c κ , d κλ ) κ,λ<α is a cylindric

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Craig Interpolation for Decidable First-Order Fragments

ten Cate,

Comer

2024

Lecture Notes in Computer Science

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We show that the guarded-negation fragment (GNFO) is, in a precise sense, the smallest extension of the guarded fragment (GFO) with Craig interpolation. In contrast, we show that the smallest extension of the two-variable fragment ($$\textrm{FO}^2 $$ FO 2 ), and of the forward fragment (FF) with Craig interpolation, is full first-order logic. Similarly, we also show that all extensions of $$\textrm{FO}^2 $$ FO 2 and of the fluted fragment (FL) with Craig interpolation are undecidable.

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Classes without the amalgamation property

Cited by 25 publications

References 5 publications

Relation Algebra with Binders

Relation Algebra with Binders

Amalgamation in finite dimensional cylindric algebras

Craig Interpolation for Decidable First-Order Fragments

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