Universal algebra

An algebraic system with an empty set of relations. A universal algebra is frequently simply called an algebra. For universal algebras the homomorphism theorem holds: If $ \phi $ is a homomorphism from one universal algebra $ A $ onto another algebra $ B $ and $ \theta $ is the kernel congruence of $ \phi $, then $ B $ is isomorphic to the quotient algebra $ A / \theta $. Every universal algebra may be decomposed into a subdirect product of subdirectly-irreducible universal algebras.

If to the basic operations of an algebra $ A $ one adjoins all derived operations, one obtains a universal algebra $ \widetilde{A} $ of larger signature. The equation $ \widetilde{A} = \widetilde{B} $ is possible even for $ A \neq B $, which leads to the notion of rational equivalence of universal algebras (cf. Variety of universal algebras).

With every universal algebra $ A $ there are associated related structures: the monoid of all endomorphisms $ \mathop{\rm End} A $, the group of all automorphisms $ \mathop{\rm Aut} A $, the lattice of all subalgebras $ \mathop{\rm Sub} A $, and the lattice of all congruences $ \mathop{\rm Con} A $. For any group $ G $ and any algebraic lattices (cf. Algebraic lattice) $ U $ and $ C $ there exists a universal algebra $ A $ such that $ G \cong \mathop{\rm Aut} A $, $ U \cong \mathop{\rm Sub} A $ and $ C \cong \mathop{\rm Con} A $( cf. [12]). However, if $ \mathop{\rm Aut} A $ is replaced by $ \mathop{\rm End} A $, the corresponding result does not hold. This kind of problem is called an abstract realization problem. An example of a solution of a concrete realization problem is: A system of subsets $ U $ of a set $ A $ coincides with $ \mathop{\rm Sub} A $ for some universal algebra with carrier $ A $ if and only if $ U $ is closed under directed unions and arbitrary intersections [11]. The abstract, as well as the concrete, realization problem can be solved for given classes of universal algebras. Universal algebras with some restriction on the related structures have been studied. E.g., one studies universal algebras with distributive or modular congruence lattices, with $ 2 $-element congruence lattices (congruence-simple universal algebras), with $ 1 $-element or $ 2 $-element lattices of subalgebras (simple universal algebras), with a commutative monoid of endomorphisms, with a $ 1 $-element group of automorphisms (rigid universal algebras), etc. A universal algebra with commuting congruences is isomorphic to the direct product of a finite number of congruence-simple algebras if and only if its congruence lattice satisfies the maximum condition, and the least upper bound of its minimal congruences is equal to the largest congruence. Universal algebras with a distributive congruence lattice and with permutable congruences (arithmetic universal algebras) admit a representation as global sections of a suitable sheaf. A study has been made to what extent a universal algebra is determined by some of its related structures. However, the majority of results of this kind concern concrete classes of universal algebras ([9]–[12], [15]).

A universal algebra is called functionally complete if every operation on its carrier belongs to the clone generated by its basic operations and by the constants. If one excludes constants, one obtains primal (or strictly functional complete) universal algebras. If all congruence-preserving operations lie in the above-mentioned clone, one obtains an affine complete universal algebra. Every functionally complete universal algebra is finite. Therefore, the requirement of finiteness is often included in the definition of these classes of universal algebras (cf. [9], [13], [14]).

The study of universal algebras was begun in the 1930's and 1940's, when the basic definitions were formulated, varieties of universal algebras were characterized and the subdirect decomposition theorem was proved (cf. [7], [8]). The earlier history of the theory of universal algebras goes back to the 19th century. The active study in this domain began in the USSR in the early 1950's (A.G. Kurosh, A.I. Mal'tsev, and their pupils). The use of methods of mathematical logic led to the consideration of algebraic systems.

The expression "universal algebra" is often used in the sense of "the theory of universal algebras" .

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In English, the phrase "universal algebra" almost always denotes "the theory of universal algebras" ; individual universal algebras (as described in the main article above) are simply called "algebras" . See also (the editorial comments to) Algebraic system.

The first occurrence in print of the phrase "universal algebra" seems to have been in [a5], although this book is less concerned with universal algebra in its modern sense than with mathematical logic.

Algebraic theories.[edit]

Universal algebra has been deepened and extended by the categorical formulation introduced by F.W. Lawvere [a2]. An algebraic theory $ T $ is defined as the category of finite copowers $ T( n) $ of an object $ T ( 1) $( with distinguished injections $ i _ {1} \dots i _ {n} $ of $ T( 1) $ to $ T( n) $). Then, a model of $ T $, or a $ T $-algebra, in a category $ C $ is a (contravariant) functor $ A : T ^ {op} \rightarrow C $ taking copowers to powers. $ A( 1) $ is the underlying object of $ A $. The standard models of $ T $ in the category of sets $ \mathop{\rm Ens} $ form precisely a variety of algebras. (These are the models $ A $ for which $ A( n) $ is the standard $ n $-th power set of $ A( 1) $.) The finitely-generated free algebras in the variety form a category equivalent to $ T $.

The operations of $ T $ are the morphisms with domain $ T( 1) $, $ \alpha : T( 1) \rightarrow T( n) $ being an $ n $-ary operation. A theory may be presented by generating operations and relations, the relations being algebraic identities; see [a4]. Morphisms of algebraic theories, or interpretations, may be defined by giving interpretations of a generating set of operations which preserve validity of defining relations. Interpretations $ T \rightarrow S $ induce algebraic functors between categories of models $ \mathop{\rm Ens} ^ {S} \rightarrow \mathop{\rm Ens} ^ {T} $; for example, the standard interpretation of Lie ring theory in associative ring theory (taking the product $ [ x, y] $ to $ xy- yx $) induces the functor which takes each associative ring to its bracket ring. Every algebraic functor has an adjoint [a2]. (In the examples, this gives the enveloping algebra.)

Co-algebras in a category $ C $ are algebras in $ C ^ {op} $. Typical algebraic theories (e.g., group theory, lattice theory) have only trivial co-algebras in $ \mathop{\rm Ens} $; see [a1].

The unrestricted infinitary extension of the notion of an algebraic theory, to a category of all copowers $ T( \aleph ) $ of an object $ T( 1) $, is called a varietal theory, but sometimes (as in [a3]) an algebraic theory; then the classical notion becomes a finitary theory. There are two equivalent formulations of varietal theories, which some authors (as in [a3]) take as definition: as monads in $ \mathop{\rm Ens} $, and as standard constructions $ ( F, U) $, $ F: A\rightarrow B $, $ U: B\rightarrow A $, with $ A= \mathop{\rm Ens} $. ( $ B $ is then the model category $ \mathop{\rm Ens} ^ {T} $.)

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