is a monadic algebra for each ,
for any ,
for all ,
for any with , and any , we have the equality
for any with and , we have the equality
where and are the abbreviations for and respectively.
Basically, the first two conditions say that the portion of the cylindric algebra is very similar to a quantifier algebra, except the domain is no longer the subsets of , but the elements of instead. The function is the algebraic abstraction of equality. Condition 3 says that is always true, condition 4 says that the proposition and its complement , where any occurrences of the variable are replaced by the variable , distinct from , is always false, while condition 5 says iff there is an such that and .
Below are some elementary properties of :
if , then
From the definition above, a cylindric algebra is a two-sorted structure, with and as the two distinct universes. However, it is often useful to view a cylindric algebra as a one-sorted structure. The way to do this is to dispense with and identify each as a unary operator on , and each as a constant in . As a result, the cylindric algebra becomes a Boolean algebra with possibly infinitely many operators:
(1) (2) (3) (4) (5) (6) (7)
Then it can be shown that is a cylindric algebra. Thus a cylindric algebra can be thought of as an “algebraization” of first order logic (with equality), much the same way as a Boolean algebra (Lindenbaum-Tarski algebra) as the algebraic counterpart of propositional logic.
- 1 P. Halmos, Algebraic Logic, Chelsea Publishing Co. New York (1962).
- 2 L. Henkin, J. D. Monk, A. Tarski, Cylindric Algebras, Part I., North-Holland, Amsterdam (1971).
- 3 J. D. Monk, Mathematical Logic, Springer, New York (1976).
- 4 B. Plotkin, Universal Algebra, Algebraic Logic, and Databases, Kluwer Academic Publishers (1994).
|Date of creation||2013-03-22 17:51:21|
|Last modified on||2013-03-22 17:51:21|
|Last modified by||CWoo (3771)|