Substitution algebras in their relation to cylindric algebras

1970 ◽  
Vol 13 (1-2) ◽  
pp. 91-96 ◽  
Author(s):  
Anne Preller
Keyword(s):  
Cylindric Algebras ◽  
Substitution Algebras

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

2014 ◽  
Vol 79 (01) ◽  
pp. 208-222 ◽  
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).

A completeness theorem for higher order logics

2000 ◽  
Vol 65 (2) ◽  
pp. 857-884 ◽  
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.

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.

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