Representation theory for polyadic algebras

Daigenault, A.; Monk, Donald

doi:10.4064/fm-52-2-151-176

Cited by 47 publications

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“…Indeed it is easily shown that, if Ai and A\ + are the polyadic algebras of a relational system with respect to two sets of variables /and /+ such that / c: /+, then Aj + is the /+-dilation of Aj. For a clue to the proof of this, see Theorem 4.4 in [7]. Then the assertion follows from remarks in §5.…”

Section: -Monomorphism and If Whenever O:a(xl)->a(x2)mentioning

confidence: 84%

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On automorphisms of polyadic algebras

Daigneault¹

1964

Trans. Amer. Math. Soc.

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Introduction. This paper belongs to the theory of polyadic algebras as developed by , but it has a bearing on the theory of models. Two central concepts are those of homogeneous(2) and of normal extensions of a polyadic algebra (see the beginning of §2 and of §6). These concepts bear some resemblance to concepts of the theory of algebraic extensions of fields; however, we have been influenced more immediately by M. Krasner's general Galois theory [33][34][35] which can be given a polyadic interpretation. Aside from the short preliminary Chapter 0, the paper is divided into sections grouped into three chapters. Each section begins by an outline of its contents. We proceed to an analysis of the main results of the paper. All polyadic algebras considered are locally finite of infinite degree.The highlights of Chapter 1 are: (i) the possibility of extending any simple extension of a (simple polyadic) algebra to a simple homogeneous and normal extension (Corollary 4.6); (ii) a first step on a Galois theory (Theorem 4.5) which, in the case of a full simple functional algebra with finite domain, takes a definite form (Theorem 4.7) closely related to Krasner's theory in that case; (iii) the existence and unicity of a+-universaI-homogeneous algebras in the sense of B. Jönsson [21; 22] in certain classes of polyadic algebras (Theorem 6.3, Theorem 6.6). The results in (iii) make use of Jönsson's work which is dependent on the continuum hypothesis (see also Theorem 5.5 which is independent of this hypothesis).Chapter II departs from the theme of automorphisms and uses little aside from Chapter 0 and §1. The main result here is Theorem 8.1 which gives a description of the functional representations of a full simple functional algebra in terms of a new generalization of the concept of reduced power.In Chapter III, we propose to show that the algebraic results of the first two chapters, when put together, embody several known model-theoretic results or new forms of such results. After a section devoted to the connections between Model theory and polyadic algebras, we give new proofs of Beth's theorem

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Section: -Monomorphism and If Whenever O:a(xl)->a(x2)mentioning

confidence: 84%

“…Let D be any filter of 0>JWK). By the generalized reduced power F(WK,X)/D we understand the (6) (fsPnhlN,-,tJN) = P(h,-,tn)IN, (7) (fNQ)*(hlN,-,tJN) = Q(h,-,QIN. [July quotient set of F(WK, X) by the equivalence relation "~" defined for elements i/, and u2 in F(WK,X) by (1) «!…”

Section: -Monomorphism and If Whenever O:a(xl)->a(x2)mentioning

confidence: 99%

On automorphisms of polyadic algebras

Daigneault¹

1964

Trans. Amer. Math. Soc.

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“…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.…”

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confidence: 94%

The class of polyadic algebras has the super amalgamation property

Ahmed¹

2010

Mathematical Logic Qtrly

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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...

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“…The references [1], [5], [6], [7], [12], [14], [15], [16] are related indirectly to the topic or are related to the applications.…”

Section: Polyadic-like Abstract Algebrasmentioning

confidence: 99%

“…The main problem, i.e., the axiomatization of the extended fields of sets, can also be approached from the representation theory of universal algebra. Infinite dimensional polyadic algebras (without equality) are representable (see [6]), but this class is a bit far from classical logic. As is known, neither cylindric algebras nor finitary polyadic equality algebras (quasi-polyadic equality algebras) can, in general, be representable in the usual universal algebraic sense.…”

Section: Introductionmentioning

confidence: 99%

The polyadic generalization of the Boolean axiomatization of fields of sets

Ferenczi

2012

Trans. Amer. Math. Soc.

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Abstract. A version of the classical representation theorem for Boolean algebras states that the fields of sets form a variety and that a possible axiomatization is the system of Boolean axioms. An important case for fields of sets occurs when the unit V is a subset of an α-power α U . Beyond the usual set operations union, intersection, and complement, new operations are needed to describe such a field of sets, e.g., the ith cylindrification C i , the constant ijth diagonal D ij , the elementary substitution [i / j] and the transposition [i, j] for all i, j < α restricted to the unit V . Here it is proven that such generalized fields of sets being closed under the above operations form a variety; further, a first order finite scheme axiomatization of this variety is presented. In the proof a crucial role is played by the existence of the operator transposition. The foregoing axiomatization is close to that of finitary polyadic equality algebras (or quasi-polyadic equality algebras).

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Representation theory for polyadic algebras

Cited by 47 publications

References 0 publications

On automorphisms of polyadic algebras

On automorphisms of polyadic algebras

The class of polyadic algebras has the super amalgamation property

The polyadic generalization of the Boolean axiomatization of fields of sets

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