Cylindric algebras and finite 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.

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A. H. Copeland Sr. Note on cylindric algebras and polyadic algebras. Michigan mathematical journal, vol. 3 (1955–1956) pp. 155–157.

Journal of Symbolic Logic ◽

10.2307/2964496 ◽

1958 ◽

Vol 23 (1) ◽

pp. 57-58

Author(s):

Paul R. Halmos

Keyword(s):

Mathematical Journal ◽

Cylindric Algebras ◽

Polyadic Algebras

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Copeland algebras

Journal of Symbolic Logic ◽

10.2307/2272408 ◽

1972 ◽

Vol 37 (4) ◽

pp. 646-656 ◽

Cited By ~ 3

Author(s):

Daniel B. Demaree

Keyword(s):

Predicate Logic ◽

Algebraic Logic ◽

Boolean Algebras ◽

Primary Concern ◽

Cylindric Algebras ◽

First Order ◽

Polyadic Algebras ◽

Algebraic Setting ◽

Finite Set ◽

Natural Way

It is well known that the laws of logic governing the sentence connectives—“and”, “or”, “not”, etc.—can be expressed by means of equations in the theory of Boolean algebras. The task of providing a similar algebraic setting for the full first-order predicate logic is the primary concern of algebraic logicians. The best-known efforts in this direction are the polyadic algebras of Halmos (cf. [2]) and the cylindric algebras of Tarski (cf. [3]), both of which may be described as Boolean algebras with infinitely many additional operations. In particular, there is a primitive operator, cκ, corresponding to each quantification, ∃υκ. In this paper we explore a version of algebraic logic conceived by A. H. Copeland, Sr., and described in [1], which has this advantage: All operators are generated from a finite set of primitive operations.Following the theory of cylindric algebras, we introduce, in the natural way, the classes of Copeland set algebras (SCpA), representable Copeland algebras (RCpA), and Copeland algebras of formulas. Playing a central role in the discussion is the set, Γ, of all equations holding in every set algebra. The reason for this is that the operations in a set algebra reflect the notion of satisfaction of a formula in a model, and hence an equation expresses the fact that two formulas are satisfied by the same sequences of objects in the model. Thus to say that an equation holds in every set algebra is to assert that a certain pair of formulas are logically equivalent.

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Projective algebra and the calculus of relations

Journal of Symbolic Logic ◽

10.2307/2271948 ◽

1978 ◽

Vol 43 (1) ◽

pp. 56-64 ◽

Cited By ~ 6

Author(s):

A. R. Bednarek ◽

S. M. Ulam

Keyword(s):

Direct Product ◽

Initial Step ◽

Point Of View ◽

Partial Product ◽

Cylindric Algebras ◽

Polyadic Algebras ◽

Algebra Of Relations ◽

Projective Algebra

In [1] there were given postulates for an abstract “projective algebra” which, in the words of the authors, represented a “modest beginning for a study of logic with quantifiers from a boolean point of view”. In [5], D. Monk observed that the study initiated in [1] was an initial step in the development of algebraic versions of logic from which have evolved the cylindric and polyadic algebras.Several years prior to the publication of [1], J. C. C. McKinsey [3] presented a set of postulates for the calculus of relations. Following the publication of [1], McKinsey [4] showed that every projective algebra is isomorphic to a subalgebra of a complete atomic projective algebra and thus, in view of the representation given in [1], every projective algebra is isomorphic to a projective algebra of subsets of a direct product, that is, to an algebra of relations.Of course there has since followed an extensive development of projective algebra resulting in the multidimensional cylindric algebras [2]. However, what appears to have been overlooked is the correspondence between the Everett–Ulam axiomatization and that of McKinsey.It is the purpose of this paper to demonstrate the above, that is, we show that given a calculus of relations as defined by McKinsey it is possible to introduce projections and a partial product so that this algebra is a projective algebra and conversely, for a certain class of projective algebras it is possible to define a multiplication so that the resulting algebra is McKinsey's calculus of relations.

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THE NEAT EMBEDDING PROBLEM FOR ALGEBRAS OTHER THAN CYLINDRIC ALGEBRAS AND FOR INFINITE DIMENSIONS

Journal of Symbolic Logic ◽

10.1017/jsl.2013.20 ◽

2014 ◽

Vol 79 (01) ◽

pp. 208-222 ◽

Cited By ~ 5

Author(s):

ROBIN HIRSCH ◽

TAREK SAYED AHMED

Keyword(s):

Embedding Problem ◽

Infinite Dimensions ◽

Cylindric Algebras ◽

First Order ◽

Polyadic Algebras ◽

Finite Set ◽

Image Position ◽

Substitution Algebras

Abstract Hirsch and Hodkinson proved, for $3 \le m < \omega $ and any $k < \omega $ , that the class $SNr_m {\bf{CA}}_{m + k + 1} $ is strictly contained in $SNr_m {\bf{CA}}_{m + k} $ and if $k \ge 1$ then the former class cannot be defined by any finite set of first-order formulas, within the latter class. We generalize this result to the following algebras of m-ary relations for which the neat reduct operator $_m $ is meaningful: polyadic algebras with or without equality and substitution algebras. We also generalize this result to allow the case where m is an infinite ordinal, using quasipolyadic algebras in place of polyadic algebras (with or without equality).

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On complete representations and minimal completions in algebraic logic, both positive and negative results

Bulletin of the Section of Logic ◽

10.18778/0138-0680.2021.17 ◽

2021 ◽

Author(s):

Tarek Sayed Ahmed

Keyword(s):

Algebraic Logic ◽

Relation Algebras ◽

Cylindric Algebras ◽

Polyadic Algebras ◽

Negative Results ◽

Complete Representations

Fix a finite ordinal $n\geq 3$ and let $\alpha$ be an arbitrary ordinal. Let $\mathsf{CA}_n$ denote the class of cylindric algebras of dimension $n$ and $\sf RA$ denote the class of relation algebras. Let $\mathbf{PA}_{\alpha}(\mathsf{PEA}_{\alpha})$ stand for the class of polyadic (equality) algebras of dimension $\alpha$. We reprove that the class $\mathsf{CRCA}_n$ of completely representable $\mathsf{CA}_n$s, and the class \(\sf CRRA$ of completely representable $\mathsf{RA}$s are not elementary, a result of Hirsch and Hodkinson. We extend this result to any variety $\sf V$ between polyadic algebras of dimension $n$ and diagonal free $\mathsf{CA}_n$s. We show that that the class of completely and strongly representable algebras in $\sf V$ is not elementary either, reproving a result of Bulian and Hodkinson. For relation algebras, we can and will, go further. We show the class $\sf CRRA$ is not closed under $\equiv_{\infty,\omega}$. In contrast, we show that given $\alpha\geq \omega$, and an atomic $\mathfrak{A}\in \mathsf{PEA}_{\alpha}$, then for any $n<\omega$, $\mathfrak{Nr}_n\A$ is a completely representable $\mathsf{PEA}_n$. We show that for any $\alpha\geq \omega$, the class of completely representable algebras in certain reducts of $\mathsf{PA}_{\alpha}$s, that happen to be varieties, is elementary. We show that for $\alpha\geq \omega$, the the class of polyadic-cylindric algebras dimension $\alpha$, introduced by Ferenczi, the completely representable algebras (slightly altering representing algebras) coincide with the atomic ones. In the last algebras cylindrifications commute only one way, in a sense weaker than full fledged commutativity of cylindrifications enjoyed by classical cylindric and polyadic algebras. Finally, we address closure under Dedekind-MacNeille completions for cylindric-like algebras of dimension $n$ and $\mathsf{PA}_{\alpha}$s for $\alpha$ an infinite ordinal, proving negative results for the first and positive ones for the second.

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Yet some more non-finite axiomatizability results for algebras of relations and ways to avoid them

Studia Scientiarum Mathematicarum Hungarica ◽

10.1556/012.2016.53.3.1341 ◽

2016 ◽

Vol 53 (3) ◽

pp. 322-378

Author(s):

Tarek Sayed Ahmed

Keyword(s):

Cylindric Algebras ◽

Finite Axiomatizability ◽

Polyadic Algebras ◽

Modified Method ◽

Complexity Results ◽

Algebras Of Relations ◽

Finitely Presented

Let α be an infinite ordinal. Let RCAα denote the variety of representable cylindric algebras of dimension α. Modifying Andréka’s methods of splitting, we show that the variety RQEAα of representable quasi-polyadic equality algebras of dimension α is not axiomatized by a set of universal formulas containing only finitely many variables over the variety RQAα of representable quasi-polyadic algebras of dimension α. This strengthens a seminal result due to Sain and Thompson, answers a question posed by Andréka, and lifts to the transfinite a result of hers proved for finite dimensions > 2. Using the modified method of splitting, we show that all known complexity results on universal axiomatizations of RCAα (proved by Andréka) transfer to universal axiomatizations of RQEAα. From such results it can be inferred that any algebraizable extension of Lω,ω is severely incomplete if we insist on Tarskian square semantics. Ways of circumventing the strong non-negative axiomatizability results hitherto obtained in the first part of the paper, such as guarding semantics, and /or expanding the signature of RQEAω by substitutions indexed by transformations coming from a finitely presented subsemigroup of (ωω, ○) containing all transpositions and replacements, are surveyed, discussed, and elaborated upon.

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Note on cylindric algebras and polyadic algebras.

10.1307/mmj/1028990028 ◽

1955 ◽

Vol 3 (2) ◽

pp. 155-157 ◽

Cited By ~ 2

Author(s):

A. H. Copeland

Keyword(s):

Cylindric Algebras ◽

Polyadic Algebras

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A completeness theorem for higher order logics

Journal of Symbolic Logic ◽

10.2307/2586575 ◽

2000 ◽

Vol 65 (2) ◽

pp. 857-884 ◽

Cited By ~ 9

Author(s):

Gábor Sági

Keyword(s):

Set Theory ◽

Representation Theorem ◽

Completeness Theorem ◽

Higher Order ◽

Algebraic Solution ◽

Algebraic Proof ◽

Relation Algebras ◽

Cylindric Algebras ◽

Strong Representation ◽

Algebraic Counterpart

AbstractHere we investigate the classes of representable directed cylindric algebras of dimension α introduced by Németi [12]. can be seen in two different ways: first, as an algebraic counterpart of higher order logics and second, as a cylindric algebraic analogue of Quasi-Projective Relation Algebras. We will give a new, “purely cylindric algebraic” proof for the following theorems of Németi: (i) is a finitely axiomatizable variety whenever α ≥ 3 is finite and (ii) one can obtain a strong representation theorem for if one chooses an appropriate (non-well-founded) set theory as foundation of mathematics. These results provide a purely cylindric algebraic solution for the Finitization Problem (in the sense of [11]) in some non-well-founded set theories.

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