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Given an algebraic category, Birkhoff's Variety Theorem gives a categorical characterization of the full subcategories whose object-class forms a variety (i.e. can be defined by equations in the sense of Model Theory).

The theorem is often stated as being of fundamental importance to Universal Algebra. As far as its importance for metamathematical questions is concerned, this does not surprise me, as it describes a connection between Model Theory and Universal Algebra. But what about its “internal” importance for Universal Algebra? Suppose we are studying a certain class of objects in some algebraic category. In how far could it be useful to know whether this class forms a variety?

My question only concerns the one implication of Birkhoff's theorem of course. It is clear that the result that varieties are closed under the taking of products, subalgebras and homomorphic images has a wide range of possible applications. But what about the converse?

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This may not be the best answer, but it is an interesting sort of application of Birkhoff's theorem.

Let's say you have some class $V$ of algebras which is defined "semantically" (e.g. as all the algebras arising from some construction), and you want to give an equivalent "syntactic" description (i.e. an axiomatization $T$). Classical examples of this include showing that the group axioms pick out the class of subgroups of permutation groups, or that the Boolean algebra axioms pick out the class of subalgebras of powerset algebras [note that I'm not claiming that Birkhoff's theorem is useful for these classical examples].

If you can show directly that $V$ is closed under the HSP operations, then you can conclude that some equational axiomatization $T^*$ exists, and this knowledge can actually help you prove that a particular equational axiomatization $T$ works.

How? Well, for example, it's clear that an algebra $A$ satisfies an equation if and only if every finitely generated subalgebra of $A$ satisfies the equation. So if you want to check that $A$ satisfies $T$ if and only if it's in $V$ (if and only if $A$ satisfies the mysterious $T^*$), then you can assume without loss that $A$ is finitely generated. And finiteness assumptions like this can be very useful.

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Reformulating part of Alex Kruckman's answer: Knowing that some class $\mathcal{A}$ of algebras has an equational presentation is useful because we then know that an algebra belongs to $\mathcal{A}$ if every finitely generated subalgebra does.

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Birkhoff's Variety Theorem is indeed of fundamental importance to Universal Algebra, notably for the stream of research it has generated, inside and outside of Universal Algebra. Here are a few examples.

The book [4] is entirely devoted to varieties of groups. See also the paper Varieties of groups by B.H. Neumann. One important result in this field states that the variety of groups generated by a finite group is finitely based. This result does not hold for arbitrary algebras (groupoids and semigroups are counter-examples). But there is a large literature on the following question:

Is it decidable whether the variety generated by a given finite algebra is finitely based?"

Birkhoff's variety theorem has also been extended in various directions. For instance, it has been extended to ordered algebras in [2] and to (pseudo)varieties of finite algebras in [1, 6] (the equations are now profinite equations) and to (pseudo)varieties of finite first-order structures in [5]. These results in turn have found important applications in the study of regular languages through Eilenberg's variety theorem [3, p. 194].

Finally, let me mention the little known but very nice book [7], which contains some interesting material on varieties and equational theories.

[1] B. Banaschewski, The Birkhoff theorem for varieties of finite algebras, Algebra Universalis 17 (1983), 360–368.

[2] S. Bloom, Varieties of ordered algebras, J. Computer System Sciences 13 (1976), 200-212.

[3] S. Eilenberg, Automata, Languages and Machines, Vol. B, Academic Press, coll. Pure and Applied Mathematics (no 59),‎ 1976, xiii+387 p.

[4] Hanna Neumann, Varieties of groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 37, Springer-Verlag, Berlin, 1967.

[5] J.-E. Pin and P. Weil, A Reiterman theorem for pseudovarieties of finite first-order structures, Algebra Universalis 35 (1996), 577–595.

[6] J. Reiterman, The Birkhoff theorem for finite algebras, Algebra Universalis 14 (1982), 1–10.

[7] Wechler, Universal Algebra for Computer Scientists, EATCS Monographs on Theoretical Computer Science 25 (1992)

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  • $\begingroup$ As the topic seems to interest you as well, let me point you to the book Algebraic Theories by Adámek, Rosicky and Vitale (published 2011). Here, Birkhoff's Theorem is treated in the following general form: Given any small category $\mathcal{T}$ with finite products, consider the category $Alg\mathcal{T}$ of finite-product-preserving functors $\mathcal{T} \to Set$. Given a set $E$ of parallel pairs of morphisms in $\mathcal{T}$, the full subcategory given by those objects of $Alg \mathcal{T}$ which merge every pair (”satisfy every equation”) of $E$, is called a variety of $Alg\mathcal{T}$... $\endgroup$ – Larry Mar 28 '16 at 8:05
  • $\begingroup$ ...Now Birkhoff's Theorem is proved in this setting, with an additional closure condition: HSP and directed unions. From your own experience, would you find generalizing to this setting useful with regard to the various extensions you mentioned? $\endgroup$ – Larry Mar 28 '16 at 8:06

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