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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.
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) $E(x,y)\leq E(y,x)$

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

$E(x,y)\wedge a=E(x,y)\wedge S(x,y)a$, 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 $x\neq y$.

$\exists(x)(E(x,y)\wedge E(x,z))=E(y,z)$ 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.
References
 1 P. Halmos, Algebraic Logic, Chelsea Publishing Co. New York (1962).
 2 B. Plotkin, Universal Algebra, Algebraic Logic, and Databases, Kluwer Academic Publishers (1994).
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