Has this notion been studied in universal algebra?

Question

In universal algebra, given an algebraic structure, there has been a lot of study of the set of equations which hold in that structure. For example, addition in the real numbers satisfies the commutative and associative identities. However, what if we have a set of equations, and are asked what set of operations satisfy all the equations of that structure. For example, let our underlying set be $\mathbb{R}$. If we have an equation $x*y=y*x$, that defines the set of all commutative binary operations $*$ on $\mathbb{R}$. Also, if we have the equation $x*(y+z)=(x*y)+(x*z)$, that defines the set of all ordered pairs $(*,+)$ of binary operations on $\mathbb{R}$ such that $*$ distributes on the left over $+$. Also, we can mix the arities of operations. For example, if we have an equation $f(x*y)=f(x)*(y)$, that defines the set of all ordered pairs $(f,*)$ where $f$ is a unary operation on $\mathbb{R}$, $*$ is a binary operation on $\mathbb{R}$, and $f$ distributes over $*$. So, my question is, has this notion been studied in some book or paper? I would like some references.

Answer 1

If you drop the implicit assumption of a fixed underlying set (e.g. $\mathbb{R}$ in what you write), then you're just describing the notion of a variety, which is one of the core notions of universal algebra. In particular, Birkhoff's HSP theorem showed that the following are equivalent, for a class of algebras (in the same language) $\mathbb{K}$:

• $\mathbb{K}$ can be axiomatized by equations, that is, $\mathbb{K}=Mod(T)$ for some equational theory $T$.

• $\mathbb{K}=\mathsf{HSP}(\mathbb{K})$, that is, every homomorphic image of a substructure of a product of elements of $\mathbb{K}$ is again an element of $\mathbb{K}$.

A crucial point here, relevant to restricting to a fixed base set, is that both small and large structures are relevant to the subclass of a variety consisting of structures of a given cardinality. For example, a class $\mathbb{F}$ of finite structures may be closed under homomorphic images, substructures, and finite products but still not be the set of finite algebras of a variety; $\mathsf{HSP}(\mathbb{F})$ may have finite elements which have to be built by "going through" an infinite element via an infinite product.

Restricting attention to a fixed base set is basically equivalent to looking at the subclass of a variety consisting of algebras of a given cardinality. We can make things more interesting by restricting attention to algebras with a fixed base set whose functions play well with a topology on that base set; this leads to a number of interesting questions and results, see Walter Taylor's paper Spaces and equations.