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Informally in mathematical logic, an algebraic theory is a theory that uses axioms stated entirely in terms of equations between terms with free variables. Inequalities and quantifiers are specifically disallowed. Sentential logic is the subset of first-order logic involving only algebraic sentences.
The notion is very close to the notion of algebraic structure, which, arguably, may be just a synonym.
Saying that a theory is algebraic is a stronger condition than saying it is elementary.