Difference between a language allowing infinitary relation/function symbols and one whose underlying logic allows infinite conjunctions/disjunctions?

Question

One description for infinitary logics (on the Wikipedia page for First-order logic) is:

Infintitary logic generalises first-order logic to allow formulas of infinite length. The most common way in which formulas can become infinite is through infinite conjunctions and disjunctions. However, it is also possible to admit generalised signatures in which functions and relation symbols are allowed to have infinite arities, or in which quantifiers can bind infinitely many variables.

I don't think I've ever seen a text go the route of admitting signatures having relation/function symbols of infinite arities. Could anyone point me to a nice instance of this being done in the literature?

I am mostly trying to get a sense of the difference between:

1. Permitting relation/function symbols of the signature to have infinite arity.
2. Permitting infinitary conjunctions/disjunctions.
3. Both (1) and (2).

Answer 1

As Noah says in the comments, very little of model theory generalizes to a setting where relation and functions symbols can have infinite arities. However, a lot of universal algebra does generalize nicely to allow function symbols of infinite arities, and there are some important classes of mathematical structures which can naturally viewed as models of infinitary algebraic theories (where infinitary here refers to the arities). A good reference is Section 1.5 of Algebraic Theories by Manes.

One starting point is the observation that any finitary algebraic theory $T$ (like the theory of groups or rings) gives rise to a monad on the category of sets such that models of $T$ are the same as algebras for the monad. If you don't know what a monad is, you can think of it as a category theoretic characterization of the situation where you have "free objects" over all sets.

Now finitary algebraic theories give rise to monads, so a natural question is whether all monads come from finitary algebraic theories. The answer is no - but it is true that every monad comes from an algebraic theory if we allow function symbols of infinite arity. In general, there won't be an upper bound on the arities required (among the infinite cardinals), so we also need to allow languages and theories of proper class size.

Having made that generalization and defined exactly what we mean by an equational axiom in a language with infinite arities, a natural question is whether all infinitary algebraic theories gives rise to monads. If the language is set sized, then the answer is yes. But for proper class sized languages (which are necessary to capture all monads), the answer is no: we again have a proper generalization. The reason is that for infinitary algebraic theories, we may not have free objects on every set. More precisely, if you try to build the free object on some set, you might end up with a proper-class sized object.

Examples:

• Complete Boolean algebras. Here the language is the usual language of Boolean algebras, extended by symbols $\bigwedge_\kappa$ and $\bigvee_\kappa$ of arity $\kappa$ for each cardinal $\kappa$. This is a proper class sized language. The class of complete Boolean algebras can be defined by an infinitary algebraic theory, but it is not the class of algebras for a monad. There is no free complete Boolean algebra on countably many generators.

• Compact Hausdorff spaces. These can be viewed as the algebras for the ultrafilter monad. Basically, for every infinite cardinal $\kappa$ and every ultrafilter $U$ on $\kappa$, we have a function symbol $f_U$ of arity $\kappa$ which takes the $U$-limit of a $\kappa$-indexed subset of the space. The relationships between these $f_U$ can be expressed as an infinitary algebraic theory. Again, we have a proper class sized language, but this time the class is monadic: there is a free compact Hausdorff spaces on any set $X$ of generators, namely the Stone-Cech compactification $\beta X$ of $X$.

• Countably complete Boolean algebras. These are like complete Boolean algebras, but we only require that countable sets have meets and joins. Again, we can axiomatize these with an infinitary equational theory, but now there is only a set sized language. So free countably complete Boolean algebraic exist, and this class is the class of algebras for a monad.