In
universal algebra
Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures in general, not specific types of algebraic structures.
For instance, rather than considering groups or rings as the object of stud ...
, 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
Term may refer to:
Language
*Terminology, context-specific nouns or compound words
**Technical term (or ''term of art''), used by specialists in a field
***Scientific terminology, used by scientists
*Term (argumentation), part of an argument in d ...
built up from variables using the operation symbols of the specified
signature
A signature (; from , "to sign") is a 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. Signatures are often, but not always, Handwriting, handwritt ...
.
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 that gives it useful properties for use in logic programming, formal specification, universal algebra and model theory. Horn clauses are ...
. A quasi-identity with ''n'' = 0 is an ordinary
identity
Identity may refer to:
* Identity document
* Identity (philosophy)
* Identity (social science)
* Identity (mathematics)
Arts and entertainment Film and television
* ''Identity'' (1987 film), an Iranian film
* ''Identity'' (2003 film), an ...
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 qua ...
References
*
Free online edition
Universal algebra
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