On amalgamation of reducts of polyadic algebras

Ahmed, Tarek Sayed

doi:10.1007/s00012-004-1807-y

Cited by 16 publications

(24 citation statements)

References 39 publications

(172 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It is also similar to the proof in [19]. However, in [17] and [19] the interpolation property is proved for countably generated free algebras in a countable signature. Our present proof deals with signatures that are necessarily uncountable (even when the dimension is ω) and so manipulations of certain cardinalities are required.…”

supporting

confidence: 58%

“…We now prove our main theorem. The proof is similar to that in [17], which studies reducts of polyadic algebras first studied by Craig [2], but is more involved due to set theoretic considerations. It is also similar to the proof in [19].…”

mentioning

confidence: 50%

“…Note that if (A, a) has a model, then a is necessarily non-zero. Due to the resemblance of the proof to that in [17] where an outline is given, the outline we present in what follows is very similar to that carried out in [17]. We apologize for the reader for this repitition, but we believe that an outline presented herein, makes the proof easier to follow.…”

mentioning

confidence: 71%

“…Our present proof deals with signatures that are necessarily uncountable (even when the dimension is ω) and so manipulations of certain cardinalities are required. Our proof uses essentially the axiom of choice, while in [17], and for that matter in [19], it is the weaker prime ideal theorem that is exploited. We first give an informal outline of the idea of the proof.…”

mentioning

confidence: 99%

“…We apologize for the reader for this repitition, but we believe that an outline presented herein, makes the proof easier to follow. We should add that although the overall idea of the proof is very similar to that in [17], there are major deviations in the details.…”

mentioning

confidence: 99%

See 4 more Smart Citations

The class of polyadic algebras has the super amalgamation property

Ahmed¹

2010

Mathematical Logic Qtrly

View full text Add to dashboard Cite

We show that for infinite ordinals α the class of polyadic algebras of dimension α has the super amalgamation property.The most well known generic examples of algebraisations of first order logic are Tarski's cylindric algebras and Halmos' polyadic algebras. Both examples are well known and are widely used. In both cases, it turns out that the locally finite algebras are representable, and this is equivalent to the completeness of first order logic. While there are cylindric algebras of infinite dimension that are not representable, Daigneault and Monk prove a strong extension of the representation theorem for locally finite polyadic algebras (due to Halmos) namely, every polyadic algebra of infinite dimension (without equality) is representable [4]. This is a point where the two theories deviate. The latter, a typical Stone-like representation theorem, reflects the fact that the notion of polyadic algebra is indeed an adequate reflection of Keisler's predicate logic (KL). KL is a proper extension of first order logic without equality, obtained when the bound on the number of variables in formulas is relaxed; and accordingly allowing the following as extra operations on formulas: Quantification on infinitely many variables and simultaneous substitution of (infinitely many) variables for variables. By the representation theorem in [4], KL is complete. Keisler gave an independent (meta-mathematical proof) of this fact.It is quite natural to ask for algebraic versions of model theoretic results, other than completeness, reflected algebraically by Stone-like representability results. Such results could concern for example interpolation theorems, or omitting types theorems Omitting types theorems for KL proves problematic, since KL is necessarily "uncountable", while omitting types theorems are very much tied to countability. However, Daigneault [3] succeeded in stating and proving versions of Beth's and Craig's theorems by proving that locally finite polyadic algebras (with and without equality) have the amalgamation property. Later Johnson removed the condition of local finiteness, proving that polyadic algebras of infinite dimension without equality have the strong amalgamation property [5]. Here we strengthen Johnson's result proving that the class of polyadic algebras has the super amalgamation property. The super amalgamation property (which follows) was introduced by Maksimova [14].Definition 1 Let W be a class of algebras each having a Boolean reduct. W has the super amalgamation property, or SUPAP for short, if for all A 0 , A 1 , A 2 ∈ W and all monomorphisms i 1 and i 2 of A 0 into A 1 , A 2 , respectively, there exists A ∈ W , a monomorphism m 1 from A 1 into A and a monomorphism m 2 from A 2 into A such that m 1 • i 1 = m 2 • i 2 , and for all x ∈ A j and for all y ∈ A k , if m j (x) ≤ m k (y), then there exists z ∈ A 0 such that x ≤ i j (z) and i k (z) ≤ y where {j, k} = {1, 2}.For an algebra A and X ⊆ A, Sg A X or simply Sg X denotes the subalgebra of A generated by X. The next definition is an algebrai...

show abstract

supporting

confidence: 58%

mentioning

confidence: 50%

mentioning

confidence: 71%

mentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

The class of polyadic algebras has the super amalgamation property

Ahmed¹

2010

Mathematical Logic Qtrly

View full text Add to dashboard Cite

show abstract

The class of infinite dimensional neat reducts of quasi-polyadic algebras is not axiomatizable

Ahmed

2006

MLQ - Math. Log. Quart.

View full text Add to dashboard Cite

SC, CA, QA and QEA denote the classes of Pinter's substitution algebras, Tarski's cylindric algebras, Halmos' quasi‐polyadic algebras and quasi‐polyadic equality algebras, respectively. Let ω ≤ α < β and let K ∈ {SC,CA,QA,QEA}. We show that the class of α ‐dimensional neat reducts of algebras in Kβ is not elementary. This solves a problem in [3]. Also our result generalizes results proved in [2] and [3]. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

show abstract

Polyadic and cylindric algebras of sentences

Amer

Ahmed

2006

Mathematical Logic Qtrly

View full text Add to dashboard Cite

In this note we give an interpretation of cylindric algebras as algebras of sentences (rather than formulas) of first order logic. We show that the isomorphism types of such algebras of sentences coincide with the class of neat reducts of cylindric algebras. Also we show how this interpretation sheds light on some recent results. This is done by likening Henkin's Neat Embedding Theorem to his celebrated completeness proof.Cylindric and quasipolyadic algebras of formulas are oriented to languages without operation symbols, see [21, Section 4.3]. All terms other than variables are ignored. Though terms may be treated algebraically afterwards, see for example [21,19,20], there seems to be some artificiality in this approach. While constructing the algebra one omits terms, and then tries to invent means of extracting terms from the algebra originally constructed without them. There have been attempts to circumvent this drawback. Indeed, algebraizing first order logic by allowing terms other than variables, i. e. treating them seriously from the very beginning, has led to the more general notion of supercylindric and superpolyadic algebras [16]. Other work in this direction (dealing with cylindric algebras of formulas with terms) includes [17]. In what follows, we also allow terms other than variables, namely individual constants, but nothing else. However, we will not be concerned with (super)cylindric nor (super)polyadic algebras of formulas with terms. Instead, we introduce cylindric algebras and quasipolyadic algebras of sentences. The novelty -to the approach adopted in the monograph [21, Section 4.3] -occurring here is twofold: Firstly, considering languages endowed with individual constants, i. e. terms other than variables. Secondly, considering cylindric and polyadic algebras of sentences and not of formulas. The trick is that we allow constants in our languages. In more detail: Let α be an ordinal. Let L be an ordinary first order language with two propositional constants and ⊥, and a list c k of individual constants of order type α. L has no operation symbols of rank > 0, but as usual, the list of variables is of order type ω. Denote by Sn L the set of all L-sentences. For ϕ ∈ Sn L and k < α let

show abstract

On amalgamation of reducts of polyadic algebras

Cited by 16 publications

References 39 publications

The class of polyadic algebras has the super amalgamation property

The class of polyadic algebras has the super amalgamation property

The class of infinite dimensional neat reducts of quasi-polyadic algebras is not axiomatizable

Polyadic and cylindric algebras of sentences

Contact Info

Product

Resources

About