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Quasi-identity

In universal algebra, a quasi-identity is an implication of the form

s<sub>1</sub> = t<sub>1</sub> ∧ … ∧ s<sub>n</sub> = t<sub>n</sub> → s = t

where s<sub>1</sub>, ..., s<sub>n</sub>, t<sub>1</sub>, ..., t<sub>n</sub>, s, and t are terms built up from variables using the operation symbols of the specified signature.

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<sub>1</sub> ≠ t<sub>1</sub> ∨ ... ∨ s<sub>n</sub> ≠ t<sub>n</sub> ∨ s = t—that is, as a definite Horn clause. A quasi-identity with n = 0 is an ordinary identity or equation, so quasi-identities are a generalization of identities.

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References