View source for Quasi-identity

''conditional identity''

Formulas of a first-order logical language of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076540/q0765401.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076540/q0765402.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076540/q0765403.png" /> denote atomic formulas of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076540/q0765404.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076540/q0765405.png" /> are terms in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076540/q0765406.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076540/q0765407.png" /> is a primitive predicate symbol. Quasi-varieties of algebraic systems are defined by quasi-identities (cf. [[Algebraic systems, quasi-variety of|Algebraic systems, quasi-variety of]]). An identity is a special case of a quasi-identity.



====Comments====
Quasi-identities are also commonly called Horn sentences.

====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></table>