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polyadic algebra (Definition)

A polyadic algebra is a quadruple % latex2html id marker 394 $ (B,V,\exists,S)$, where % latex2html id marker 396 $ (B,V,\exists)$ is a quantifier algebra, and $ S$ is a function from the set of functions on $ V$ to the set of endomorphisms on the Boolean algebra $ B$, in other words

$\displaystyle S:V^V\to \operatorname{End}(B)$
such that
  1. $ S(1_V)=1_B$,
  2. $ S(f\circ g)=S(f)\circ S(g)$,
  3. % latex2html id marker 410 $ S(f)\circ \exists(I)=S(g)\circ \exists(I)$ if $ f(V-I)=g(V-I)$,
  4. % latex2html id marker 414 $ \exists(I)\circ S(f) = S(f)\circ \exists(f^{-1}(I))$ if $ f$ is one-to-one when restricted to $ f^{-1}(I)$.

Explanation of notations: $ 1_V,1_B$ are identity functions on $ V,B$ respectively; $ f,g$ are functions on $ V$, and $ I$ is a subset of $ V$. The circle $ \circ$ represents functional compositions.

The degree and local finiteness of a polyadic algebra are defined as the degree and local finiteness of the underlying quantifier algebra.

Heuristically, the function $ S$ can be thought of as changes to propositional functions due to a “substitution” of variables (elements of $ V$). Let us see some examples. Let $ V=\lbrace x_0,x_1,\ldots \rbrace$ be a countably indexed set of variables. For any propositional function $ \varphi$, define $ S(f)(\varphi)$ to be the propositional function $ \varphi_1$ obtained by replacing each variable $ x$ that occurs in it by $ f(x)$. Below are two examples illustrating how $ S(f)$ changes propositional functions:

  • Let $ f:V \to V$ be the function given by $ f(x_0)=f(x_1)=x_0$ and $ f(x_i)=x_{i+1}$ for all $ i>1$. If $ \varphi$ is the propositional function $ x_0^2-x_1+x_2/x_3$, then $ S(f)(\varphi)$ is the propositional function $ x_0^2-x_0+x_3/x_4$.
  • Let $ f:V\to V$ be the function given by $ f(x_0)=x_2$, and $ f(x_i)=x_i$ for all $ i\ne 0$. Then the propositional function “ % latex2html id marker 476 $ \exists x_0,x_1,x_2 (x_0\ne x_1 \wedge x_1\ne x_2 \wedge x_2\ne x_0)$” becomes “ % latex2html id marker 478 $ \exists x_2,x_1,x_2 (x_2\ne x_1 \wedge x_1\ne x_2 \wedge x_2\ne x_2)$” under $ S(f)$.
It is not hard to see from the examples above that $ S(f)$ respects Boolean operations $ \wedge$ and $ '$, which is why we want to make $ S(f)$ an endomorphism on $ B$. Furthermore, the four conditions above can be interpreted as
  1. if there are no substitutions of variables, then there should be no corresponding changes to the propositional functions
  2. applying substitutions $ f\circ g$ of varaibles in a propositional function $ \varphi$ should have the same effect as applying substitutions $ g$ of variables in $ \varphi$, followed by substitutions $ f$ of variables in $ S(g)(\varphi)$
  3. a substitution $ f$ of variables should have no effect to a propositional function beginning with % latex2html id marker 506 $ \exists$ if every variable bound by % latex2html id marker 508 $ \exists$ is fixed by $ f$. For example, if we replace $ f$ in the second example above by $ f(x_3)=x_2$ and $ f(x_i)=x_i$ otherwise, then
    % latex2html id marker 518 $\displaystyle \lq\lq \exists x_0,x_1,x_2 (x_0\ne x_1 \wedge x_1\ne x_2 \wedge x_2\ne x_0)''$
    is unchanged by $ S(f)$, since $ x_0,x_1,x_2$ are all fixed by $ f$.
  4. Let % latex2html id marker 526 $ \varphi=\exists I \psi(I,J)$ be a propositional function, where $ I,J$ are sets of variables with $ I$ bound by % latex2html id marker 532 $ \exists$ and $ J$ free. If no two variables $ I$ get mapped to the same variable, and no free variable (in $ J$) becomes bound (in $ f(I)$) under the substitution, then % latex2html id marker 542 $ \exists I \psi(f(I),f(J))$ and % latex2html id marker 544 $ \exists f(I) \psi(f(I),f(J))$ are semantically the same, which is exactly the statement in the condition.

Remarks.

  • Paul Halmos first introduced the notion of polyadic algebras. In his Algebraic Logic, a compilation of articles on the subject, he called a function on the set $ V$ of variables a transformation, and the triple $ (B,V,S)$ satisfying the first two conditions above a transformation algebra. Therefore, a polyadic algebra is a quadruple % latex2html id marker 550 $ (B,V,\exists,S)$ where % latex2html id marker 552 $ (B,V,\exists)$ is a quantifier algebra and $ (B,V,S)$ is a transformation algebra, such that conditions 3 and 4 above are satisfied.
  • The notion of polyadic algebras generalizes the notion of monadic algebras. Indeed, a monadic algebra is a polyadic algebra where $ V$ is a singleton.
  • Just as a Lindenbaum-Tarski algebra is the algebraic counterpart of a classical propositional logic, a polyadic algebra is the algebraic counterpart of a classical first order logic without equality. A variant of the polyadic algebra is what is known as a cylindric algebra, which algebratizes a classical first order logic with equality.

Bibliography

1
P. Halmos, Algebraic Logic, Chelsea Publishing Co. New York (1962).
2
B. Plotkin, Universal Algebra, Algebraic Logic, and Databases, Kluwer Academic Publishers (1994).



"polyadic algebra" is owned by CWoo.
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See Also: quantifier algebra, monadic algebra, cylindric algebra

Also defines:  transformation algebra

Attachments:
polyadic algebra with equality (Definition) by CWoo
example of polyadic algebra (Example) by CWoo
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Cross-references: cylindric algebra, equality, classical first order logic, propositional logic, Lindenbaum-Tarski algebra, singleton, monadic algebras, transformation, logic, algebraic, free variable, fixed, bound, operations, Boolean, occur ins, variables, propositional functions, degree, compositions, functional, represents, circle, subset, identity functions, restricted, one-to-one, Boolean algebra, endomorphisms, function, quantifier algebra
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This is version 10 of polyadic algebra, born on 2008-02-22, modified 2008-02-26.
Object id is 10315, canonical name is PolyadicAlgebra.
Accessed 704 times total.

Classification:
AMS MSC: 03G15 (Mathematical logic and foundations :: Algebraic logic :: Cylindric and polyadic algebras; relation algebras)
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