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Difference between revisions of "Quasi-identity"

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''conditional identity''
 
''conditional identity''
  
Formulas of a first-order logical language of the form
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Formulae of a first-order logical language of the form
 
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$$
<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>
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(\forall x_1,\ldots,x_n)\,(A_1 \wedge \cdots \wedge A_p \rightarrow A)
 
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$$
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
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where $A_1,\ldots,A_p$ and $A$ denote atomic formulae of the form
 
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$$
<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>
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f = g\ \ \text{or}\ \ P(\alpha_1,\ldots,\alpha_m)
 
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$$
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.
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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====
 
====Comments====
Quasi-identities are also commonly called Horn sentences.
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Quasi-identities are also commonly called ''Horn sentences'' or ''Horn clauses'': see [[Horn clauses, theory of]].
  
 
====References====
 
====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>
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<table>
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<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>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  P.M. Cohn,  "Universal algebra" , Reidel  (1981)  pp. 235</TD></TR>
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</table>
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Revision as of 21:39, 20 October 2016

conditional identity

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.

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
How to Cite This Entry:
Quasi-identity. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quasi-identity&oldid=39459
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article