example of cylindric algebra
Consider , the three-dimensional Euclidean space, and
Thus is the closed unit ball, centered at the origin . Project onto the - plane, so its image is
Taking its preimage, we get a cylinder
has the following properties:
Furthermore, it can be characterized as follows
is called the cylindrification of with respect to the variable . It is easy to see that the characterization above permits us to generalize the notion of cylindrification to any subset of , with respect to any of the three variables . We have in addition to (1) above the following properties:
where and .
Property (2) is obvious. To see Property (3), it is enough to assume (for the other cases follow similarly). First let . Then there is an such that and , which means there is an such that . Since , we have that , and since , we have that as well. This shows one inclusion. Now let , then there is an such that . But also, so . To see Property (4), it is enough to assume and . Let . Then there is an such that , and so there is an such that . This implies that , which implies that . So . The other inclusion then follows immediately.
Next, we define the diagonal set
with respect to and . This is just the plane whose projection onto the - plane is the line . We may define a total of nine possible diagonal sets where . However, there are in fact four distinct diagonal sets, since
where . For any subset , set . For instance, .
To see Property (7), we may assume and . Suppose . Then there is such that , which implies that , or that . On the other hand, there is such that , which implies , or that , a contradiction. To see Property (8), we may assume . If , then there is such that . So and . Therefore, . On the other hand, for any , , and so as well.
Write . It is easy to see that is a Boolean algebra with operations . Next define by where , and by . Then Properties (1), (2), and (3) show that is a monadic algebra, and Properties (4), (5), (7), and (8) show that is cylindric. ∎
Example 2 (Cylindric Set Algebras).
Example 1 above may be generalized. Let be sets, and set . For any subset and any , define the cylindrification of by
and the diagonal set by
Now, define and by and .
is a cylindric algebra, called a cylindric set algebra.
The proof of this can be easily derived based on the discussion in Example 1, and is left for the reader as an exercise.
Remark. For more examples of cylindric algebras, see the second reference below.
- 1 L. Henkin, J. D. Monk, A. Tarski, Cylindric Algebras, Part I., North-Holland, Amsterdam (1971).
- 2 J. D. Monk, Connections Between Combinatorial Theory and Algebraic Logic, Studies in Algebraic Logic, The Mathematical Association of America, (1974).
- 3 J. D. Monk, Mathematical Logic, Springer, New York (1976).
- 4 B. Plotkin, Universal Algebra, Algebraic Logic, and Databases, Kluwer Academic Publishers (1994).
|Title||example of cylindric algebra|
|Date of creation||2013-03-22 17:52:26|
|Last modified on||2013-03-22 17:52:26|
|Last modified by||CWoo (3771)|
|Defines||cylindric set algebra|