example of polyadic algebra

Recall that the canonical example of a monadic algebra is that of a functional monadic algebra, which is a pair (B,) such that B is the set of all functionsMathworldPlanetmath from a non-empty set X to a Boolean algebraMathworldPlanetmath A such that, for each fB, the supremumMathworldPlanetmathPlanetmath and the infimumMathworldPlanetmath of f(X) exist, and is a function on B that maps each element f to f, a constant element whose range is a singleton consisting of the supremum of f(X).

The canonical example of a polyadic algebra is an extensionPlanetmathPlanetmath (generalizationPlanetmathPlanetmath) of a functional monadic algebra, known as the functional polyadic algebra. Instead of looking at functions from X to A, we look at functions from XI (where I is some set), the I-fold cartesian power of X, to A. In this entry, an element xXI is written as a sequence of elements of A: (xi)iI where xiA, or (xi) for short.

Before constructing the functional polyadic algebra based on the sets X,I and the Boolean algebra A, we first introduce the following notations:

  • for any JI and xXI, define the subset (of XI)

    [x]J:={yXIxi=yi for every iJ},
  • for any function τ:II and any f:XIA, define the function fτ from XI to A, given by


Now, let B be the set of all functions from XI to A such that

  1. 1.

    for every fB, every JI and every xXI, the arbitrary join



    Before stating the next condition, we introduce, for each fB, a function fJ:XIA as follows:


    Now, we are ready for the next condition:

  2. 2.

    if fB, then fJB,

  3. 3.

    if fB, then fτB for τ:II.

Note that if A were a complete Boolean algebra, we can take B to be AXI, the set of all functions from XI to A.

Next, define :P(I)BB by (J)(f)=fJ, and let S be the semigroup of functions on I (with functionalPlanetmathPlanetmathPlanetmath compositions as multiplicationsPlanetmathPlanetmath), then we call the quadruple (B,I,,S) the functional polyadic algebra for the triple (A,X,I).

Remarks. Let (B,I,,S) be the functional polyadic algebra for (A,X,I).

  • (B,I,,S) is a polyadic algebra. The proof of this is not difficult, but involved, and can be found in the reference below.

  • If I is a singleton, then (B,I,,S) can be identified with the functional monadic algebra (B,) for (A,X), for S is just I, and XI is just X.

  • If I is , then (B,I,,S) can be identified with the Boolean algebra A, for S= and XI is a singleton, and hence the set of functions from XI to A is identified with A.


Title example of polyadic algebra
Canonical name ExampleOfPolyadicAlgebra
Date of creation 2013-03-22 17:53:20
Last modified on 2013-03-22 17:53:20
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 15
Author CWoo (3771)
Entry type Example
Classification msc 03G15
Defines functional polyadic algebra