Step by step – Building representations in algebraic logic

Hirsch, Robin; Hodkinson, Ian

doi:10.2307/2275740

Cited by 44 publications

(49 citation statements)

References 24 publications

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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%

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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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“…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%

“…In the proof of the Main Theorem (Theorem 3.1) we follow the step-by-step method (see [12] or [2]) applied, for example, in Andréka and Thompson's proof for the Resek-Thompson theorem (Trans. Amer.…”

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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“…It will be necessary to have to hand axioms for the representable relation algebras, and here we recall the axioms obtained by games. We assume familiarity with basic aspects of relation algebras; see, e.g., [29,15] The following definition is from [15, definition 7.12] (see also [14]; here we have changed the notation slightly). A strategy for a player in G n (A) is a set of rules giving that player a non-empty set of permissible moves in any situation.…”

Section: Games On Relation Algebrasmentioning

confidence: 99%

“…His proof used relation algebra reducts of cylindric algebras and is unpublished, but was reported briefly in [30, theorem 2.12]. The first full published proof is in [28], using 'n-dimensional relation algebras', and a proof using saturation was given in [14] and [15, theorem 3.36]. It can also be proved using a general phenomenon in the duality theory of BAOs, namely, that a variety of BAOs is canonical if it is generated by a class of relational structures that is closed under ultraproducts [12].…”

Section: Introductionmentioning

confidence: 99%

Canonical varieties with no canonical axiomatisation

Hodkinson

Venema

2004

Trans. Amer. Math. Soc.

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Abstract. We give a simple example of a variety V of modal algebras that is canonical but cannot be axiomatised by canonical equations or first-order sentences. We then show that the variety RRA of representable relation algebras, although canonical, has no canonical axiomatisation. Indeed, we show that every axiomatisation of these varieties involves infinitely many noncanonical sentences.Using probabilistic methods of Erdős, we construct an infinite sequence G 0 , G 1 , . . . of finite graphs with arbitrarily large chromatic number, such that each G n is a bounded morphic image of G n+1 and has no odd cycles of length at most n. The inverse limit of the sequence is a graph with no odd cycles, and hence is 2-colourable. It follows that a modal algebra (respectively, a relation algebra) obtained from the G n satisfies arbitrarily many axioms from a certain axiomatisation of V (RRA), while its canonical extension satisfies only a bounded number of them. First-order compactness will now establish that V (RRA) has no canonical axiomatisation. A variant of this argument shows that all axiomatisations of these classes have infinitely many non-canonical sentences.

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“…We will use two-player games in the characterization, and translate the existence of a winning strategy for one of the players into a set of first-order axioms; thus, we find, for an arbitrary class of the form SCmV, an axiomatization with strong intuitive content. Similar techniques were used to construct axiomatizations in [16,6,8,7,14]. The method is implicitly used in the much earlier [12], although games are not mentioned per se.…”

Section: Introductionmentioning

confidence: 99%