Operations of higher or unspecified arity are usually denoted by function symbols, with the arguments placed in parentheses and separated by commas, like f(x,y,z) or f(x1,...,xn).
is an ordered sequence of natural numbers representing the arity of the operations of the algebra.
A collection of algebraic structures defined by identities is called a variety or equational class.
Restricting one's study to varieties rules out: The study of equational classes can be seen as a special branch of model theory, typically dealing with structures having operations only (i.e. the type can have symbols for functions but not for relations other than equality), and in which the language used to talk about these structures uses equations only.
One advantage of this restriction is that the structures studied in universal algebra can be defined in any category that has finite products.
This definition of a group does not immediately fit the point of view of universal algebra, because the axioms of the identity element and inversion are not stated purely in terms of equational laws which hold universally "for all ..." elements, but also involve the existential quantifier "there exists ...".
For example, when defining a group object in category theory, where the object in question may not be a set, one must use equational laws (which make sense in general categories), rather than quantified laws (which refer to individual elements).
For example, in a topological group, the inverse must not only exist element-wise, but must give a continuous mapping (a morphism).
Then there are three basic constructions in universal algebra: homomorphic image, subalgebra, and product.
In addition to its unifying approach, universal algebra also gives deep theorems and important examples and counterexamples.
Smith puts it, "What looks messy and complicated in a particular framework may turn out to be simple and obvious in the proper general one."
Universal algebra provides a natural language for the constraint satisfaction problem (CSP).
CSP refers to an important class of computational problems where, given a relational algebra A and an existential sentence
The dichotomy conjecture (proved in April 2017) states that if A is a finite algebra, then CSPA is either P or NP-complete.
A more recent development in category theory is operad theory – an operad is a set of operations, similar to a universal algebra, but restricted in that equations are only allowed between expressions with the variables, with no duplication or omission of variables allowed.
Whitehead credits William Rowan Hamilton and Augustus De Morgan as originators of the subject matter, and James Joseph Sylvester with coining the term itself.
In a review Alexander Macfarlane wrote: "The main idea of the work is not unification of the several methods, nor generalization of ordinary algebra so as to include them, but rather the comparative study of their several structures.
Whitehead's early work sought to unify quaternions (due to Hamilton), Grassmann's Ausdehnungslehre, and Boole's algebra of logic.
Work on the subject was minimal until the early 1930s, when Garrett Birkhoff and Øystein Ore began publishing on universal algebras.
Developments in metamathematics and category theory in the 1940s and 1950s furthered the field, particularly the work of Abraham Robinson, Alfred Tarski, Andrzej Mostowski, and their students.
Although the development of mathematical logic had made applications to algebra possible, they came about slowly; results published by Anatoly Maltsev in the 1940s went unnoticed because of the war.
Starting with William Lawvere's thesis in 1963, techniques from category theory have become important in universal algebra.