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.

Comments

Quasi-identities are also commonly called Horn sentences or Horn clauses: see Horn clauses, theory of.

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

References

[a1]	A. Horn, "On sentences which are true of direct unions of algebras" J. Symbol. Logic , 16 (1951) pp. 14–21
[a2]	P.M. Cohn, "Universal algebra" , Reidel (1981) pp. 235
[b1]	A.I. Mal'tsev, "Algebraic systems" , Springer (1973) (Translated from Russian)