> But some sets of data and some operations on them that fulfill some formally stated requirements are just an abstract algebra, aren't they?
Not quite. A variety of algebras (which is usually what people have in mind when they talk about "algebraic structures" in general) is a collection of operations with equational laws, meaning that they're of the form "for all x_0, x_1, ... (however many variables you need), expression_1(x_0, x_1, ...) = expression_2(x_0, x_1, ...)", where the expressions are built up out of your operators.
Fields are the classic example of a structure studied by algebraists which is nonetheless not a variety of algebras: division isn't described by equational laws, because it's defined for everything except zero. This makes fields much harder to work with than e.g. groups or rings, in both math and programming.
Not quite. A variety of algebras (which is usually what people have in mind when they talk about "algebraic structures" in general) is a collection of operations with equational laws, meaning that they're of the form "for all x_0, x_1, ... (however many variables you need), expression_1(x_0, x_1, ...) = expression_2(x_0, x_1, ...)", where the expressions are built up out of your operators.
Fields are the classic example of a structure studied by algebraists which is nonetheless not a variety of algebras: division isn't described by equational laws, because it's defined for everything except zero. This makes fields much harder to work with than e.g. groups or rings, in both math and programming.