Isomorphic but not lower base-isomorphic cylindric set algebras

B. Biró; S. Shelah

doi:10.2307/2274576

Isomorphic but not lower base-isomorphic cylindric set algebras

Journal of Symbolic Logic ◽

10.2307/2274576 ◽

1988 ◽

Vol 53 (3) ◽

pp. 846-853 ◽

Cited By ~ 5

Author(s):

B. Biró ◽

S. Shelah

Keyword(s):

Model Theory ◽

Algebraic Logic ◽

Algebraic Model ◽

Locally Finite ◽

Lower Base ◽

Cylindric Set Algebras

AbstractThis paper belongs to cylindric-algebraic model theory understood in the sense of algebraic logic. We show the existence of isomorphic but not lower base-isomorphic cylindric set algebras. These algebras are regular and locally finite. This solves a problem raised in [N 83] which was implicitly present also in [HMTAN 81]. This result implies that a theorem of Vaught for prime models of countable languages does not continue to hold for languages of any greater power.

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Algebraic Model Theory

10.1007/978-94-015-8923-9 ◽

1997 ◽

Cited By ~ 1

Keyword(s):

Model Theory ◽

Algebraic Model

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STONE SPACE OF CYLINDRIC ALGEBRAS AND TOPOLOGICAL MODEL SPACES

Journal of Symbolic Logic ◽

10.1017/jsl.2016.11 ◽

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.

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Intelligently deciphering unintelligible designs: algorithmic algebraic model checking in systems biology

Journal of The Royal Society Interface ◽

10.1098/rsif.2008.0546 ◽

2009 ◽

Vol 6 (36) ◽

pp. 575-597 ◽

Cited By ~ 6

Author(s):

Bud Mishra

Keyword(s):

Systems Biology ◽

Model Checking ◽

Model Theory ◽

Historical Context ◽

Algebraic Model ◽

The Subject ◽

And Algebra

Systems biology, as a subject, has captured the imagination of both biologists and systems scientists alike. But what is it? This review provides one researcher's somewhat idiosyncratic view of the subject, but also aims to persuade young scientists to examine the possible evolution of this subject in a rich historical context. In particular, one may wish to read this review to envision a subject built out of a consilience of many interesting concepts from systems sciences, logic and model theory, and algebra, culminating in novel tools, techniques and theories that can reveal deep principles in biology—seen beyond mere observations. A particular focus in this review is on approaches embedded in an embryonic program, dubbed ‘algorithmic algebraic model checking’, and its powers and limitations.

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On the number of generators of cylindric algebras

Journal of Symbolic Logic ◽

10.2307/2273976 ◽

1985 ◽

Vol 50 (4) ◽

pp. 865-873

Author(s):

H. Andréka ◽

I. Németi

Keyword(s):

Model Theory ◽

Order Logic ◽

First Order Logic ◽

Single Element ◽

Abstract Model ◽

Finitely Generated ◽

Cylindric Algebras ◽

Abstract Model Theory ◽

First Order ◽

Cylindric Set Algebras

The theory of cylindric algebras (CA's) is the algebraic theory of first order logics. Several ideas about logic are easier to formulate in the frame of CA-theory. Such are e.g. some concepts of abstract model theory (cf. [1] and [10]–[12]) as well as ideas about relationships between several axiomatic theories of different similarity types (cf. [4] and [10]). In contrast with the relationship between Boolean algebras and classical propositional logic, CA's correspond not only to classical first order logic but also to several other ones. Hence CA-theoretic results contain more information than their counterparts in first order logic. For more about this see [1], [3], [5], [9], [10] and [12].Here we shall use the notation and concepts of the monographs Henkin-Monk-Tarski [7] and [8]. ω denotes the set of natural numbers. CAα denotes the class of all cylindric algebras of dimension α; by “a CAα” we shall understand an element of the class CAα. The class Dcα ⊆ CAα was defined in [7]. Note that Dcα = 0 for α ∈ ω. The classes Wsα, and Csα were defined in 1.1.1 of [8], p. 4. They are called the classes of all weak cylindric set algebras, regular cylindric set algebras and cylindric set algebras respectively. It is proved in [8] (I.7.13, I.1.9) that ⊆ CAα. (These inclusions are proper by 7.3.7, 1.4.3 and 1.5.3 of [8].)It was proved in 2.3.22 and 2.3.23 of [7] that every simple, finitely generated Dcα is generated by a single element. This is the algebraic counterpart of a property of first order logics (cf. 2.3.23 of [7]). The question arose: for which simple CAα's does “finitely generated” imply “generated by a single element” (see p. 291 and Problem 2.3 in [7]). In terms of abstract model theory this amounts to asking the question: For which logics does the property described in 2.3.23 of [7] hold? This property is roughly the following. In any maximal theory any finite set of concepts is definable in terms of a single concept. The connection with CA-theory is that maximal theories correspond to simple CA's (the elements of which are the concepts of the original logic) and definability corresponds to generation.

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