polyadic algebra with equality

Let $A=(B,V,\exists ,S)$ be a polyadic algebra. An equality predicate on $A$ is a function $E:V\times V\to B$ such that

1. $S(f)\circ E(x,y)=E(f(x),f(y))$ for any $f:V\to V$ and any $x,y\in V$

2. $E(x,x)=1$ for every $x\in V$, and

3. $E(x,y)\wedge a\le S(x/y)a$, where $a\in B$, and $(x/y)$ denotes the function $V\to V$ that maps $x$ to $y$, and constant everywhere else.

Heuristically, we can interpret the conditions above as follows:

1. if $x=y$ and if we replace $x$ by, say $x_1$, and $y$ by $y_1$, then $x_1=y_1$.

2. $x=x$ for every variable $x$

3. if we have a propositional function $a$ that is true, and $x=y$, then the proposition obtained from $a$ by replacing all occurrences of $x$ by $y$ is also true.

The second condition is also known as the reflexive property of the equality predicate $E$, and the third is known as the substitutive property of $E$

A polyadic algebra with equality is a pair $(A,E)$ where $A$ is a polyadic algebra and $E$ is an equality predicate on $A$. Paul Halmos introduced this concept and called this simply an equality algebra.

Below are some basic properties of the equality predicate $E$ in an equality algebra $(A,E)$:

• (symmetric property) $E(x,y)\le E(y,x)$

• (transitive property) $E(x,y)\wedge E(y,z)\le E(x,z)$

• $E(x,y)\wedge a=E(x,y)\wedge S(x,y)a$, where $(x,y)$ in the $S$ is the transposition on $V$ that swaps $x$ and $y$ and leaves everything else fixed.

• if a variable $x\in V$ is not in the support of $a\in A$, then $a=\exists (x)(E(x,y)\wedge S(y/x)a)$.

• $\exists (x)(E(x,y)\wedge a)\wedge \exists (x)(E(x,y)\wedge a^{\prime })=0$ for all $a\in A$ and all $x,y\in V$ whenever $x\ne y$.

• $\exists (x)(E(x,y)\wedge E(x,z))=E(y,z)$ for all $x,y,z\in V$ where $x\notin \{y,z\}$.

Remarks

• The degree and local finiteness of a polyadic algebra $(A,E)$ are defined as the degree and the local finiteness and degree of its underlying polyadic algebra $A$.

• It can be shown that every locally finite polyadic algebra of infinite degree can be embedded (as a polyadic subalgebra) in a locally finite polyadic algebra with equality of infinite degree.

• Like cylindric algebras, polyadic algebras with equality is an attempt at “converting” a first order logic (with equality) into algebraic form, so that the logic can be studied using algebraic means.

Title	polyadic algebra with equality
Canonical name	PolyadicAlgebraWithEquality
Date of creation	2013-03-22 17:51:37
Last modified on	2013-03-22 17:51:37
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	10
Author	CWoo (3771)
Entry type	Definition
Classification	msc 03G15
Synonym	equality algebra
Related topic	CylindricAlgebra
Defines	equality predicate
Defines	substitutive
Defines	reflexive
Defines	symmetric
Defines	transitive