In
universal algebra
Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures.
For instance, rather than take particular groups as the object of study, ...
, a quasi-identity is an implication of the form
:''s''
1 = ''t''
1 ∧ … ∧ ''s''
''n'' = ''t''
''n'' → ''s'' = ''t''
where ''s''
1, ..., ''s''
''n'', ''t''
1, ..., ''t''
''n'', ''s'', and ''t'' are
terms built up from variables using the operation symbols of the specified
signature
A signature (; from la, signare, "to sign") is a handwritten (and often stylized) depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. The writer of a ...
.
A quasi-identity amounts to a conditional equation for which the conditions themselves are equations. Alternatively, it can be seen as a disjunction of inequations and one equation ''s''
1 ≠ ''t''
1 ∨ ... ∨ ''s''
''n'' ≠ ''t''
''n'' ∨ ''s'' = ''t''—that is, as a definite
Horn clause In mathematical logic and logic programming, a Horn clause is a logical formula of a particular rule-like form which gives it useful properties for use in logic programming, formal specification, and model theory. Horn clauses are named for the log ...
. A quasi-identity with ''n'' = 0 is an ordinary
identity
Identity may refer to:
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* Identity (philosophy)
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* Identity (mathematics)
Arts and entertainment Film and television
* ''Identity'' (1987 film), an Iranian film
* ''Identity'' (2003 film), ...
or equation, so quasi-identities are a generalization of identities.
See also
*
Quasivariety In mathematics, a quasivariety is a class of algebraic structures generalizing the notion of variety by allowing equational conditions on the axioms defining the class.
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Definition
A ''trivial algebra'' contains just one element. A quasiv ...
References
*
Free online edition
Universal algebra
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