Difference between revisions of "Quasi-identity"
From Encyclopedia of Mathematics
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| − | ''conditional 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> | ||
| − | + | {{TEX|done}} | |
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Latest revision as of 07:40, 21 October 2016
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) |
How to Cite This Entry:
Quasi-identity. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quasi-identity&oldid=14576
Quasi-identity. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quasi-identity&oldid=14576
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article