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.

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.

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 $a=\exists(x) (E(x,y)\wedge S(y/x)a)$ .
• $\exists(x)(E(x,y)\wedge a)\wedge \exists(x)(E(x,y)\wedge a')=0$ 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 $x\notin \lbrace y,z\rbrace$ .

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.

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).