Is it a requirement for algebraic structures, when studying universal algebra, to satisfy axioms? The reason I ask is because algebraic structures are only defined by a underlying set, a signature, and a interpretation function. There is no mention of algebraic structures having to satisfy any axioms. If the answer is no, how can you study classes of algebraic structures that are fields, for example, if some of them do not satisfy the field axioms but still share the same signature of that of a field.
2026-03-25 21:46:51.1774475211
Are algebraic structures required to satisfy axioms?
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Every structure does satisfy some statements and fails to satisfy the negations of the statements that it satisfies.
Every set of statements defines some class of structures: those that satisfy the statements in the set.
Some sets of axioms and some classes of structures are worth more attention than others.
Not all algebraic structures satisfy the field axioms. If by "required" you mean: must some particular axioms be satisfied in order that an algebraic structure be an algebraic structure? Then the answer is no, unless you mean that having an underlying set or some operations amounts to satisfying "axioms". But that's no reason why one cannot study the class of all algebraic structures that do satisfy some specified axioms.