A non-representable cylindric algebra with pairing functions

1986 ◽  
Vol 22 (2-3) ◽  
pp. 117-119 ◽  
Author(s):  
I. N�meti
Keyword(s):  
Cylindric Algebra

Nonfinitizability of classes of representable cylindric algebras

1969 ◽  
Vol 34 (3) ◽  
pp. 331-343 ◽  
Author(s):  
J. Donald Monk
Keyword(s):  
Predicate Logic ◽  
Infinite Dimension ◽  
Fundamental Result ◽  
Cylindric Algebra ◽  
Cylindric Algebras ◽  
Locally Finite ◽  
First Order ◽  
Alfred Tarski ◽  
Sentential Logic ◽  
Algebraic Counterpart

Cylindric algebras were introduced by Alfred Tarski about 1952 to provide an algebraic analysis of (first-order) predicate logic. With each cylindric algebra one can, in fact, associate a certain, in general infinitary, predicate logic; for locally finite cylindric algebras of infinite dimension the associated predicate logics are finitary. As with Boolean algebras and sentential logic, the algebraic counterpart of completeness is representability. Tarski proved the fundamental result that every locally finite cylindric algebra of infinite dimension is representable.

STONE SPACE OF CYLINDRIC ALGEBRAS AND TOPOLOGICAL MODEL SPACES

2016 ◽  
Vol 81 (3) ◽  
pp. 1069-1086
Author(s):  
CHARLES C. PINTER
Keyword(s):  
Boolean Algebra ◽  
Model Theory ◽  
Representation Theorem ◽  
Stone Space ◽  
Equivalence Relations ◽  
Cylindric Algebra ◽  
Cylindric Algebras ◽  
Locally Finite ◽  
Abstract Model Theory ◽  
Image Position

AbstractThe Stone representation theorem was a milestone for the understanding of Boolean algebras. From Stone’s theorem, every Boolean algebra is representable as a field of sets with a topological structure. By means of this, the structural elements of any Boolean algebra, as well as the relations between them, are represented geometrically and can be clearly visualized. It is no different for cylindric algebras: Suppose that ${\frak A}$ is a cylindric algebra and ${\cal S}$ is the Stone space of its Boolean part. (Among the elements of the Boolean part are the diagonal elements.) It is known that with nothing more than a family of equivalence relations on ${\cal S}$ to represent quantifiers, ${\cal S}$ represents the full cylindric structure just as the Stone space alone represents the Boolean structure. ${\cal S}$ with this structure is called a cylindric space.Many assertions about cylindric algebras can be stated in terms of elementary topological properties of ${\cal S}$. Moreover, points of ${\cal S}$ may be construed as models, and on that construal ${\cal S}$ is called a model space. Certain relations between points on this space turn out to be morphisms between models, and the space of models with these relations hints at the possibility of an “abstract” model theory. With these ideas, a point-set version of model theory is proposed, in the spirit of pointless topology or category theory, in which the central insight is to treat the semantic objects (models) homologously with the corresponding syntactic objects so they reside together in the same space.It is shown that there is a new, purely algebraic way of introducing constants in cylindric algebras, leading to a simplified proof of the representation theorem for locally finite cylindric algebras. Simple rich algebras emerge as homomorphic images of cylindric algebras. The topological version of this theorem is especially interesting: The Stone space of every locally finite cylindric algebra ${\frak A}$ can be partitioned into subspaces which are the Stone spaces of all the simple rich homomorphic images of ${\frak A}$. Each of these images completely determines a model of ${\frak A}$, and all denumerable models of ${\frak A}$ appear in this representation.The Stone space ${\cal S}$ of every cylindric algebra can likewise be partitioned into closed sets which are duals of all the types in ${\frak A}$. This fact yields new insights into miscellaneous results in the model theory of saturated models.

scholarly journals Concerning some cylindric algebra versions of the downward Löwenheim-Skolem theorem.

1988 ◽  
Vol 29 (3) ◽  
pp. 332-344 ◽  
Author(s):  
Ildikó Sain
Keyword(s):  
Cylindric Algebra ◽  
Skolem Theorem

Cylindric algebra equation solver

1990 ◽  
pp. 645-646
Author(s):  
Frank M. Brown ◽  
Carlos Araya
Keyword(s):  
Cylindric Algebra

Nonfinitizability of classes of representable polyadic algebras

1969 ◽  
Vol 34 (3) ◽  
pp. 344-352 ◽  
Author(s):  
James S. Johnson
Keyword(s):  
Predicate Logic ◽  
Free Algebras ◽  
Cylindric Algebra ◽  
Cylindric Algebras ◽  
Polyadic Algebras

The notion of polyadic algebra was introduced by Halmos to reflect algebraically the predicate logic without equality. Later Halmos enriched the study with the introduction of the notion of equality. These algebras are very closely related to the cylindric algebras of Tarski. The notion of diagonal free cylindric algebra predates that of cylindric algebra and is also due to Tarski. The theory of diagonal free algebras forms an important fragment of the theories of polyadic and cylindric algebras.

The equational theory of CA3 is undecidable

1980 ◽  
Vol 45 (2) ◽  
pp. 311-316 ◽  
Author(s):  
Roger Maddux
Keyword(s):  
Equational Theory ◽  
Cylindric Algebra ◽  
Relation Algebras ◽  
Cylindric Algebras ◽  
3 Dimensional ◽  
First Order ◽  
Equational Theories ◽  
Order Language ◽  
Theoretic Proof

There is no algorithm for determining whether or not an equation is true in every 3-dimensional cylindric algebra. This theorem completes the solution to the problem of finding those values of α and β for which the equational theories of CAα and RCAβ are undecidable. (CAα and RCAβ are the classes of α-dimensional cylindric algebras and representable β-dimensional cylindric algebras. See [4] for definitions.) This problem was considered in [3]. It was known that RCA0 = CA0 and RCA1 = CA1 and that the equational theories of these classes are decidable. Tarski had shown that the equational theory of relation algebras is undecidable and, by utilizing connections between relation algebras and cylindric algebras, had also shown that the equational theories of CAα and RCAβ are undecidable whenever 4 ≤ α and 3 ≤ β. (Tarski's argument also applies to some varieties K ⊆ RCAβ with 3 ≤ β and to any variety K such that RCAα ⊆ K ⊆ CAα and 4 ≤ α.)Thus the only cases left open in 1961 were CA2, RCA2 and CA3. Shortly there-after Henkin proved, in one of Tarski's seminars at Berkeley, that the equational theory of CA2 is decidable, and Scott proved that the set of valid sentences in a first-order language with only two variables is recursive [11]. (For a more model-theoretic proof of Scott's theorem see [9].) Scott's result is equivalent to the decidability of the equational theory of RCA2, so the only case left open was CA3.

scholarly journals Strongly representable atom structures of cylindric algebras

2009 ◽  
Vol 74 (3) ◽  
pp. 811-828 ◽  
Author(s):  
Robin Hirsch ◽  
Ian Hodkinson
Keyword(s):  
Chromatic Number ◽  
Cylindric Algebra ◽  
Cylindric Algebras ◽  
Free Graph ◽  
Complex Algebra

AbstractA 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 , and prove, for infinite Γ, that 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 non-principal ultraproduct ΠDΓk whose chromatic number is just two. It follows that is strongly representable (each k < ω) but is not.

Rectangular games

1998 ◽  
Vol 63 (4) ◽  
pp. 1549-1564 ◽  
Author(s):  
Yde Venema
Keyword(s):  
Order Logic ◽  
First Order Logic ◽  
Cylindric Algebra ◽  
First Order

AbstractWe prove that every rectangularly dense diagonal-free cylindric algebra is representable. As a corollary, we give finite, sound and complete axiomatizations for the finite-variable fragments of first order logic without equality and for multi-dimensional modal S5-logic.

Free Boolean algebras with closure operators and a conjecture of Henkin, Monk, and Tarski

2001 ◽  
Vol 38 (1-4) ◽  
pp. 273-278
Author(s):  
I. Németi ◽  
J. X. Madarász
Keyword(s):  
Boolean Algebras ◽  
Cylindric Algebra ◽  
Cylindric Algebras ◽  
Closure Operators ◽  
Finite Dimensional

We characterize the finite-dimensional elements of a free cylindric algebra. This solves Problem 2.10 in [Henkin, Monk, Tarski: Cylindric Algebras, North-Holland, 1971 and 1985]. We generalize the characterization to quasi-varieties of Boolean algebras with op- erators in place of cylindric algebras.

Cylindric algebras with terms

1990 ◽  
Vol 55 (2) ◽  
pp. 854-866 ◽  
Author(s):  
Norman Feldman
Keyword(s):  
Predicate Logic ◽  
Equivalence Classes ◽  
Cylindric Algebra ◽  
Cylindric Algebras ◽  
First Order ◽  
Order Language ◽  
First Order Predicate Logic

In this paper we discuss cylindric algebras with terms. The setting is two—sorted algebras—one sort for terms and one for Boolean elements. As with cylindric algebras, a cylindric algebra with terms has its roots in first order predicate logic [HMT1].Let Σ be a set of sentences in a first order language with terms, equality and variables u0,u1,u2, …, Define a relation ≡Σ on Fm, the set of formulas, by φ ≡Σθ if and only if Σ ⊢ φ ↔ θ, and on Tm, the set of terms, by τ ≡Σσ if and only if Σ ⊢ τ ≈ σ. The operations +, ·, cκ, 0, 1 are defined as usual on equivalence classes. Define , where is σ with τ substituted for all occurrences of uκ. That the operation *κ, for κ < α, is well defined follows from the first order axioms of equality. Let vκ = [uκ]. To establish the link between terms and Booleans, define operations oκ as follows: , where φ' is a variant of φ such that uκ is free for τ in φ′ and is φ′ with τ substituted for all free occurrences of uκ in φ′. From the first order axioms it follows that oκ, for κ < α, is well defined. Finally, instead of diagonal elements, we define a Boolean-valued operation on terms as follows: [τ] e [σ] = [τ ≈ σ].

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