# Quasi-identity

conditional identity, Horn clause

Formulae of a first-order logical language of the form $$(\forall x_1,\ldots,x_n)\,(A_1 \wedge \cdots \wedge A_p \rightarrow A)$$ where $A_1,\ldots,A_p$ and $A$ denote atomic formulae of the form $$f = g\ \ \text{or}\ \ P(\alpha_1,\ldots,\alpha_m)$$ where $f,g,\alpha_1,\ldots,\alpha_m$ are terms in $x_1,\ldots,x_n$ and $P$ is a primitive predicate symbol. Quasi-varieties of algebraic systems are defined by quasi-identities (cf. Algebraic systems, quasi-variety of). An identity is a special case of a quasi-identity.

In this context, an "identity" is a formula $$(\forall x_1,\ldots,x_n)\,( A) \ .$$