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17
Finite Schematizable Algebraic Logic
, 1997
"... In this work, we attempt to alleviate three (more or less) equivalent negative results. These are (i) nonaxiomatizability (by any nite schema) of the valid formula schemas of rst order logic, (ii) nonaxiomatizability (by nite schema) of any propositional logic equivalent with classical rst ..."
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In this work, we attempt to alleviate three (more or less) equivalent negative results. These are (i) nonaxiomatizability (by any nite schema) of the valid formula schemas of rst order logic, (ii) nonaxiomatizability (by nite schema) of any propositional logic equivalent with classical rst order logic (i.e., modal logic of quanti cation and substitution), and (iii) nonaxiomatizability (by nite schema) of the class of representable cylindric algebras (i.e., of the algebraic counterpart of rst order logic). Here we present two nite schema axiomatizable classes of algebras that contain, as a reduct, the class of representable quasipolyadic algebras and the class of representable cylindric algebras, respectively. We establish positive results in the direction of nitary algebraization of rst order logic without equality as well as that with equality. Finally, we will indicate how these constructions can be applied to turn negative results (i), (ii) above to positive ones.
On the Search for a Finitizable Algebraization of First Order Logic
, 2000
"... We give an algebraic version of first order logic without equality in which the class of representable algebras forms a nitely based equational class. Further, the representables are dened in terms of set algebras, and all operations of the latter are permutation invariant. The algebraic form of thi ..."
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We give an algebraic version of first order logic without equality in which the class of representable algebras forms a nitely based equational class. Further, the representables are dened in terms of set algebras, and all operations of the latter are permutation invariant. The algebraic form of this result is Theorem 1.1 (a concrete version of which is given by Theorems 2.8 and 4.2), while its logical form is Corollary 5.2. For first order logic with equality we give a result weaker than the one for rst order logic without equality. Namely, in this case  instead of finitely axiomatizing the corresponding class of all representable algebras  we finitely axiomatize only the equational theory of that class. See Subsection 6.1, especially Remark 6.6 there. The proof of Theorem 1.1 is elaborated in Sections 3 and 4. These sections contain theorems which are interesting of their own rights, too, e.g. Theorem 4.2 is a purely semigroup theoretic result. Cf. also "Further main results" in the
Strongly representable atom structures of cylindric algebras
, 2007
"... 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 ≥ ..."
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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 ndimensional 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, loopfree graph Γ, we construct an ndimensional 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 Γk (k < ω) with infinite chromatic number, but having a nonprincipal ultraproduct � D Γk whose chromatic number is just two. It follows that E(Γk) is strongly representable (each k < ω) but � D E(Γk) is not. 1
Strongly representable atom structures of relation algebras
 PROC. AMER. MATH. SOC
, 2001
"... A relation algebra atom structure α is said to be strongly representable if all atomic relation algebras with that atom structure are representable. This is equivalent to saying that the complex algebra Cm α is a representable relation algebra. We show that the class of all strongly representable r ..."
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A relation algebra atom structure α is said to be strongly representable if all atomic relation algebras with that atom structure are representable. This is equivalent to saying that the complex algebra Cm α is a representable relation algebra. We show that the class of all strongly representable relation algebra atom structures is not closed under ultraproducts and is therefore not elementary. This answers a question of Maddux (1982). Our proof is based on the following construction. From an arbitrary undirected, loopfree graph Γ, we construct a relation algebra atom structure α(Γ) and prove, for infinite Γ, that α(Γ) is strongly representable if and only if the chromatic number of Γ is infinite. A construction of Erdös shows that there are graphs Γr (r <ω) with infinite chromatic number, with a nonprincipal ultraproduct � D Γr whose chromatic number is just two. It follows that α(Γr) is strongly representable (each r<ω) but � D (α(Γr)) is not.
Algebras of relations of various ranks, some current trends and applications
 Journal on Relational Methods in Computer Science 1 (2004), 2749. 26 H. Andréka, I. Németi and I. Sain, Algebraic Logic. In: Handbook of Philosophical Logic, Vol 2
, 2001
"... Abstract. Here the emphasis is on the main pillars of Tarskian structuralist approach to logic: relation algebras, cylindric algebras, polyadic algebras, and Boolean algebras with operators. We also tried to highlight the recent renaissance of these areas and their fusion with new trends related to ..."
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Abstract. Here the emphasis is on the main pillars of Tarskian structuralist approach to logic: relation algebras, cylindric algebras, polyadic algebras, and Boolean algebras with operators. We also tried to highlight the recent renaissance of these areas and their fusion with new trends related to logic, like the guarded fragment or dynamic logic. Tarskian algebraic logic is far too broad and too fruitful and prolific by now to be covered in a short paper like this. Therefore the overview part of the paper is rather incomplete, we had to omit important directions as well as important results. Hopefully, this incompleteness will be alleviated by the accompanying paper of Tarek Sayed Ahmed [81]. The structuralist approach to a branch of learning aims for separating out the really essential things in the phenomena being studied abstracting from the accidental wrappings or details (called in computer science “syntactic sugar”). As a result of this, eventually one associates to the original phenomena (or systems) being studied streamlined elegant mathematical structures. These streamlined structures can be algebras in the sense of universal algebra, or other kinds of elegant well understood mathematical structures like e.g. spacetime geometries
Nonfinitely axiomatisable twodimensional modal logics
, 2011
"... We show the first examples of recursively enumerable (even decidable) twodimensional products of finitely axiomatisable modal logics that are not finitely axiomatisable. In particular, we show that any axiomatisation of some bimodal logics that are determined by classes of product frames with linea ..."
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We show the first examples of recursively enumerable (even decidable) twodimensional products of finitely axiomatisable modal logics that are not finitely axiomatisable. In particular, we show that any axiomatisation of some bimodal logics that are determined by classes of product frames with linearly ordered first components must be infinite in two senses: It should contain infinitely many propositional variables, and formulas of arbitrarily large modal nestingdepth. 1
Representable Cylindric Algebras and ManyDimensional Modal Logics
, 2010
"... The equationally expressible properties of the cylindrifications and the diagonals in finitedimensional representable cylindric algebras can be divided into two groups: (i) ‘Onedimensional ’ properties describing individual cylindrifications. These can be fully characterised by finitely many equat ..."
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The equationally expressible properties of the cylindrifications and the diagonals in finitedimensional representable cylindric algebras can be divided into two groups: (i) ‘Onedimensional ’ properties describing individual cylindrifications. These can be fully characterised by finitely many equations saying that each ci, for i < n, is a normal (ci0 = 0), additive (ci(x+y) = cix+ciy) and complemented closure operator: x ≤ cix cicix ≤ cix ci(−cix) ≤ −cix. (1) (ii) ‘Dimensionconnecting ’ properties, that is, equations describing the diagonals and interaction between different cylindrifications and/or diagonals. These properties are much harder to describe completely, and there are many results in the literature on their complexity. The main aim of this chapter is to study generalisations of (i) while keeping (ii) as unchanged as possible. In other words, we would like to analyse how much of the complexity of RCAn is due to its ‘manydimensional ’ character and how much of it
On axiomatising products of Kripke frames, part II
 Advances in Modal Logic, Volume 7
, 2008
"... abstract. We generalise some results of [7, 5] and show that if L is an αmodal logic (for some ordinal α ≥ 3) such that (i) L contains the product logic Kα and (ii) the product of αmany trees of depth one and with arbitrary large finite branching is a frame for L, then any axiomatisation of L must ..."
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abstract. We generalise some results of [7, 5] and show that if L is an αmodal logic (for some ordinal α ≥ 3) such that (i) L contains the product logic Kα and (ii) the product of αmany trees of depth one and with arbitrary large finite branching is a frame for L, then any axiomatisation of L must contain infinitely many propositional variables. As a consequence we obtain that product logics like Kα, K4α, S4α, GLα, and Grzα cannot be axiomatised using finitely many propositional variables, whenever α ≥ 3.
Bare canonicity of representable cylindric and polyadic algebras
"... We show that for finite n ≥ 3, every firstorder axiomatisation of the varieties of representable ndimensional cylindric algebras, diagonalfree cylindric algebras, polyadic algebras, and polyadic equality algebras contains an infinite number of noncanonical formulas. We also show that the class o ..."
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We show that for finite n ≥ 3, every firstorder axiomatisation of the varieties of representable ndimensional cylindric algebras, diagonalfree cylindric algebras, polyadic algebras, and polyadic equality algebras contains an infinite number of noncanonical formulas. We also show that the class of structures for each of these varieties is nonelementary. The proofs employ algebras derived from random graphs.