Relational clone

Definition

Let $A$ be a nonempty set. Let $\Gamma$ be a set of finitary relations on $A$. A primitive positive formula over $\Gamma$ is a formula of the form $$\exists y_1 \cdots \exists y_k \, \bigl(\varphi_1 \wedge \cdots \wedge \varphi_r\bigr).$$ Here each $\varphi_j$ is an equation $z_1 = z_2$ or an atomic formula $R(z_1, \dots, z_m)$ with $R \in \Gamma$ of arity $m$. The $z_i$ are variables. A relation $S \subseteq A^p$ is primitive positive definable from $\Gamma$ if $S = \set{(a_1, \dots, a_p) \in A^p : \psi(a_1, \dots, a_p)}$. Here $\psi(x_1, \dots, x_p)$ is a primitive positive formula over $\Gamma$ with free variables $x_1, \dots, x_p$. A relational clone on $A$ is a set $\Gamma$ of finitary relations on $A$ such that:

1. $\Gamma$ contains the binary equality relation $\set{(a, a) : a \in A}$.
2. $\Gamma$ contains the empty relation of each arity.
3. Every relation that is primitive positive definable from $\Gamma$ belongs to $\Gamma$.