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

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