Omitting types for finite variable fragments and complete representations of algebras

Andréka, Hajnal; Németi, István; Ahmed, Tarek Sayed

doi:10.2178/jsl/1208358743

Cited by 23 publications

(48 citation statements)

References 38 publications

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“…Recall that an algebra A is simple if for any algebra B of the same signature, any homomorphism h : A → B is either trivial (i.e., h(x) = h(y) for all x, y ∈ A ) or one-one. 2 Lemma 5.1. C is simple, as is any subalgebra of C .…”

Section: and Suppose That {Amentioning

confidence: 99%

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Strongly representable atom structures of cylindric algebras

Hirsch¹,

Hodkinson²

2009

J. symb. log.

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Abstract. A cylindric algebra atom structure is said to be strongly representable if all atomic cylindric algebras with that atom structure are representable. This is equivalent to saying that the full complex algebra of the atom structure is a representable cylindric algebra. We show that for any finite n ≥ 3, the class of all strongly representable n-dimensional cylindric algebra atom structures is not closed under ultraproducts and is therefore not elementary.Our proof is based on the following construction. From an arbitrary undirected, loop-free graph Γ, we construct an n-dimensional atom structure E (Γ), and prove, for infinite Γ, that E (Γ) is a strongly representable cylindric algebra atom structure if and only if the chromatic number of Γ is infinite. A construction of Erdős shows that there are graphs Γ k (k < ) with infinite chromatic number, but having a non-principal ultraproductIntroduction. This paper is broadly about algebras of α-ary relations, for an ordinal α. An α-ary relation is a set of ordered α-tuples of elements of some base set, and an algebra of α-ary relations will consist of a set of α-ary relations, endowed with various operations. These operations include the boolean union and complement and constants denoting the empty relation and the maximum or 'unit' relation, and the algebra will be a boolean algebra under these operations. But there will also be additional operations that make use of the relational form of the elements of the algebra. Various choices of these operations can be made. The 'cylindric' approach, first taken by Alfred Tarski and his students Louise Chin and Frederick Thompson in the late 1940s, gives us cylindric set algebras, which have since been studied extensively, e.g., in [10,8, 9]. These algebras include constants called diagonal elements, which are like equality, and unary functions called cylindrifications, which are like existential quantification. For finite α, the algebras are closely related to first-order logic with α variables. But relation symbols in first-order logic have finite arity, so for infinite α, the algebraic approach, which can handle relations of any arity up to α, goes further.Roughly speaking, the class RCA α of 'representable α-dimensional cylindric algebras' is the closure under isomorphism of the class of all algebras of relations as just described. Quite a lot of work has gone into characterising RCA α . Tarski proved in [22] that it is a variety: it can be axiomatised by equations. Explicit finite sets of

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Section: and Suppose That {Amentioning

confidence: 99%

“…We assume some knowledge of basic boolean notions such as atoms and ultrafilters. For those seeking more details of the topics considered here, we suggest [8,9,19,21,2], or for some parts, [12].…”

mentioning

confidence: 99%

Strongly representable atom structures of cylindric algebras

Hirsch¹,

Hodkinson²

2009

J. symb. log.

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“…A different version of the above theorem is proved in [2]. We note that our construction quite easily leads to the (new) fact that the Omitting Types Theorem fails for finite first order definable extensions of finite variable fragments of first order logic studied in [5] and [16] as long as the number of variables is > 2.…”

Section: Omitting Types For Finite Variable Fragmentsmentioning

confidence: 88%

“…That is, we obtain our desired atom structures in one blow. (This is also done in [2]. But we should add that the construction in [2] is completely different and much more complex.…”

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Weakly representable atom structures that are not strongly representable, with an application to first order logic

Ahmed¹

2008

Mathematical Logic Qtrly

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Let n > 2. A weakly representable relation algebra that is not strongly representable is constructed. It is proved that the set of all n by n basic matrices forms a cylindric basis that is also a weakly but not a strongly representable atom structure. This gives an example of a binary generated atomic representable cylindric algebra with no complete representation. An application to first order logic is given.

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On notions of representability for cylindric‐polyadic algebras, and a solution to the finitizability problem for quantifier logics with equality

Ahmed

2015

Mathematical Logic Qtrly

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We consider countable so‐called rich subsemigroups of (ωω,∘); each such semigroup T gives a variety CPEAT that is axiomatizable by a finite schema of equations taken in a countable subsignature of that of ω‐dimensional cylindric‐polyadic algebras with equality where substitutions are restricted to maps in T. It is shown that for any such T, A∈ CPEA T if and only if frakturA is representable as a concrete set algebra of ω‐ary relations. The operations in the signature are set‐theoretically interpreted like in polyadic equality set algebras, but such operations are relativized to a union of cartesian spaces that are not necessarily disjoint. This is a form of guarding semantics. We show that CPEAT is canonical and atom‐canonical. Imposing an extra condition on T, we prove that atomic algebras in CPEAT are completely representable and that CPEAT has the super amalgamation property. If T is rich and finitely represented, it is shown that CPEAT is term definitionally equivalent to a finitely axiomatizable Sahlqvist variety. Such semigroups exist. This can be regarded as a solution to the central finitizability problem in algebraic logic for first order logic with equality if we do not insist on full fledged commutativity of quantifiers. The finite dimensional case is approached from the view point of guarded and clique guarded (relativized) semantics of fragments of first order logic using finitely many variables. Both positive and negative results are presented.

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Omitting types for finite variable fragments and complete representations of algebras

Cited by 23 publications

References 38 publications

Strongly representable atom structures of cylindric algebras

Strongly representable atom structures of cylindric algebras

Weakly representable atom structures that are not strongly representable, with an application to first order logic

On notions of representability for cylindric‐polyadic algebras, and a solution to the finitizability problem for quantifier logics with equality

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