Locally Finite Functional Polyadic Algebra with Terms

: In 1956 P.R.Halmos [3] introduced polyadic algebra to express first order logic algebraically. In this work we extend polyadic algebra to include terms as


INTRODUCTION
The extension of Boolean algebra to several variable functional Boolean algebra with certain operators is capable of expressing polyadic logic [1] in an algebraic form called polyadic algebra [3] The operators correspond to the usual existential and universal quantifiers. The main issue of this paper is the study of terms as an embedded part of the system of polyadic algebra. Our study case is the locally finite functional polyadic algebra over a countable number of variables. The importance of functional case stems from the fact that any locally finite polyadic algebra is isomorphic to a functional polyadic algebra [3]. At the end of the paper a suggestion for the general case of locally finite polyadic algebra with terms is made.

General definition [3]:
Suppose that B is a complete Boolean algebra. An existential quantifier on B is a mapping : BB such that i) (0)  0 ii) a(a) for any aB. iii) (a (b)) (a) (b) for any a,bB.
(B,) is called monadic algebra. Let I be a set usually countable to index the variables. A mapping not necessarily one to one or onto from I into itself is called a transformation.

Let
Denote the set of all endomorphisms on B by End(B) and the set of all quantifiers on B by Qant(B). A polyadic algebra is such that for any 6) if  is one to one on  -1 (J). The cardinal number I is called the degree of the algebra. If I = n we get the so called n-adic algebra. If I =  then 2 I  and   id. Therefore we get only the identity quantifier () = id . Thus 0-adic algebra is just the Boolean algebra B. If I = 1 then   id. Therefore there are only two quantifiers () and (I).
Thus 1-adic algebra is the monadic algebra.

Local Finiteness:
A simplified version of polyadic algebra has been provided for a locally finite polyadic algebra [2]. It is done as follows: . It is well known that any finite transformation on I is a finite product of replacements [3]. Denote is called independent of J if (J)(a)  a. We say that J is support a if a is independent of I  J i.e (I J)(a)  a. A polyadic algebra is called locally finite if for any aB there is a finite subset a J of I such that a J supports a. In particular when I is finite B is locally finite. Now any locally finite polyadic algebra B is characterized as follows: Theorem 1 For any aB there is a finite subset a J of I such that the system satisfies the following conditions:

FUNCTIONAL POLYADIC ALGEBRA
Le be a complete Boolean algebra, where A is an algebra of type F. For p,qB A define , and 0,1 pointwise as follows: for any aA. We have Proposition 2 [2] is a functional Boolean algebra. Proposition 3 [2] is a monadic algebra, where is given by Proposition 4 [2] is a polyadic algebra, where and Now, let . Define for any and . We have Proposition 5 is a polyadic algebra Proof

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Volume 2 The algebra is locally finite, and hence it satisfies the following conditions: They are satisfied as follows: We have

Bulletin of Mathematical Sciences and Applications Vol. 2 63
Proposition 6 is a polyadic algebra Proof The algebra is also locally finite and hence it satisfies the following conditions They are satisfied as follows:

TERMS
Let X be a set of variables and F be a type of algebra. The set T(X) of terms of type F over X is the smallest set such that 64 Volume 2 The set T(X) can be transformed into an algebra [5]. The term algebra T(X) of type F over X has as its universe the set T(X) and the fundamental operations satisfy : Now, consider the functional polyadic algebra A term t defines a function Thus obviously we have Proposition 7 Let be a term on the algebra A. By using finite compositions of product, projection, identity and /or constant functions on x 1 , x 2 ,..., x n the term t can be transformed uniquely to a function, say again t: is a function also for i =1,2,..., k.
Not that proposition 7 is a special case of proposition 8. Finally, consider the countable indexing set I 1,2,3,...of variables on A. A term t in a function involving a finite number of such variables. As above t induces a mapping given as follows:

Proposition 9
Proof Now it is possible to introduce a locally finite functional polyadic algebra with terms by where B is a complete Boolean algebra, A any algebra of type F, I The first six conditions above may be replaced by the simpler five conditions of locally finite case as before.