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 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 or equation, so quasi-identities are a generalization of identities.
See also
*
Quasivariety
References
*
Free online edition
Universal algebra
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