Difference between abstract algebra and universal algebra

Solution 1:

Three typical questions of abstract algebra (with well-known answers):

• What is the structure of the group of units $\mathbb{Z}[\sqrt{2}]^*$?
• Is every subgroup of a free group on $n$ generators free?
• What are the maximal ideals of $\mathbb{C}[x_1,\dotsc,x_n]$?

Three typical questions in universal algebra:

• If a variety has a nontrivial algebra, does it have a nontrivial simple algebra?
• How can we detect if a variety is congruence-permutable?
• For which varieties of finite type is the theory of the finite algebras decidable?

In some sense, universal algebra is abstract (abstract algebra): Objects of study are varieties (i.e. the class of all sets equipped with certain operations which satisfy some rules). In abstract algebra one studies objects of a fixed variety. Many constructions and notions of abstract algebra can be generalized within universal algebra.

Solution 2:

I will attempt to answer the question without assuming knowledge of what a variety is.

In abstract algebra we define the algebra we are studying as a set with specific operations and specific rules or axioms. For example, if we are studying groups, we define a group as a tuple $ \langle G, \cdot, ^{-1}, 1 \rangle $ where $ G $ is a set, $ \cdot $ is a binary operation on $ G $, $ ^{-1} $ is a unary operation on $ G $ and 1 is a constant. The rules are

1. For all $ x,y,z \in G $, $ x \cdot ( y \cdot z ) = ( x \cdot y ) \cdot z $
2. For all $ x \in G $, $ x^{-1} \cdot x = 1 $ and $ x \cdot x^{-1} = 1 $
3. etc.

In contrast, in universal algebra we begin by defining an algebra as a tuple $ \langle A, \rho \rangle $, where $ A $ is a set with some arbitrary set of operations $ \rho $. When comparing or manipulating algebras we then normally restrict our algebras to be similar, i.e. have the same type, that is each of their operations should have the same order, for example in groups it is binary, unary and constant. However, we do not always specify the type.

This added level of abstraction allows us to ask different questions about algebras. For instance we can ask: "if we have a class of similar algebras, under what conditions do the algebras satisfy the same equations?" Or "under what conditions can these equations be reduced to a finite number?" Or "how can we find all algebras that satisfy the same equations as a given one."

If a class of similar algebras are closed with respect to satisfying the same equations (as apposed to something more general like sentences) we call it a variety. These are one of the main objects of study in universal algebra. For instance the class of all groups is trivially a variety because they are defined by the equations above.