example of polyadic algebra with equality
Recall that given a triple $(A,I,X)$ where $A$ is a Boolean algebra^{}, $I$ and $X\ne \mathrm{\varnothing}$ are sets. we can construct a polyadic algebra $(B,I,\exists ,S)$ called the functional polyadic algebra for $(A,I,X)$. In this entry, we will construct an example of a polyadic algebra with equality called the functional polyadic algebra with equality from $(B,I,\exists ,S)$.
We start with a simpler structure^{}. Let $B$ be an arbitrary Boolean algebra, $I$ and $X\ne \mathrm{\varnothing}$ are sets. Let $Y={X}^{I}$, the set of all $I$-indexed $X$-valued sequences, and $Z={B}^{Y}$, the set of all functions from $Y$ to $B$. Call the function $e:I\times I\to Z$ the functional equality associated with $(B,I,X)$, if for each $i,j\in I$, $e(i,j)$ is the function defined by
$$e(i,j)(x):=\{\begin{array}{cc}1\hfill & \text{if}{x}_{i}={x}_{j},\hfill \\ 0\hfill & \text{otherwise.}\hfill \end{array}$$ |
The quadruple $(B,I,X,e)$ is called a functional equality algebra.
Now, $B$ will have the additional structure of being a polyadic algebra. Start with a Boolean algebra $A$, and let $I$ and $X$ be defined as in the last paragraph. Then, as stated above in the first paragraph, and illustrated in here (http://planetmath.org/ExampleOfPolyadicAlgebra), $(B,I,\exists ,S)$ is a polyadic algebra (called the functional polyadic algebra for $(A,I,X)$). Using the $B$ just constructed, the quadruple $(B,I,X,e)$ is a functional equality algebra, and is called the functional polyadic algebra with equality for $(A,I,X)$.
It is not hard to show that $e$ is an equality predicate on $C=(B,I,\exists ,S)$, and as a result $(C,e)$ is a polyadic algebra with equality.
References
- 1 P. Halmos, Algebraic Logic, Chelsea Publishing Co. New York (1962).
Title | example of polyadic algebra with equality |
---|---|
Canonical name | ExampleOfPolyadicAlgebraWithEquality |
Date of creation | 2013-03-22 17:55:24 |
Last modified on | 2013-03-22 17:55:24 |
Owner | CWoo (3771) |
Last modified by | CWoo (3771) |
Numerical id | 5 |
Author | CWoo (3771) |
Entry type | Example |
Classification | msc 03G15 |
Defines | functional equality algebra |
Defines | functional equality |
Defines | functional polyadic algebra with equality |