# Can we represent all algebraic structures in First-Order logic?

Can we represent all algebraic structures in First-Order logic?

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What, no background? Lame. –  Ｊ. Ｍ. Sep 26 '10 at 14:32

The subject of universal algebra is the study of algebraic structures as classes of first-order structures in languages having only function symbols as opposed to relation symbols. This subject is regarded by many as much closer to algebra than the rest of model theory, allowing one to combine methods from model theory with a stronger category-theoretic perspective. A central concern are the classes of algebras, called varieties, defined by a system of algebraic equations.

For example, Birkoff's theorem states that a class of algebraic structures is a variety of algebras if and only if it is closed under homomorphisms, subalgebras and products.

A similar corresponding theorem in first order logic is due to Kiesler and Shelah: two structures are elementary equivalent if and only if they have a common ultrapower, and one can use this to characterize which collections of algebras form an elementary class.

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I am not sure I understand your question. If, by definition, an algebraic structure consists of a set $A\ne\emptyset$ (or a family of such sets) together with a collection of finitary functions on $A$ (functions $f:A^n\to A$ or even ${\mathbf f}:A^n\to A^m$), finitary relations (subsets of $A^k$), and constants (elements of $A$), then 'algebraic structures' are precisely the structures that first-order concerns itself with (you have to use a multi-sorted version of first-order if you want to appropriately work in the setting of "families of sets").

Any such structure can be axiomatized in a first-order language as long as you are willing to allow enough function, relation, and constant symbols into your language.

Of course, the Lowenheim-Skolem theorems will prevent you from uniquely characterizing infinite structures unless you impose additional conditions that are not first-order (like "completeness", for the reals).

But perhaps you mean something else. If you are interested in families of such structures, which is more natural, then the answer is no in general (simple groups, for example). And many interesting "structures" that appear in algebra are second (or higher) order in nature, and typically first-order falls short to capture them explicitly.

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