Universal algebra Algebraic structure

algebraic structures defined through different configurations of axioms. universal algebra abstractly studies such objects. 1 major dichotomy between structures axiomatized entirely identities , structures not. if axioms defining class of algebras identities, class of objects variety (not confused algebraic variety in sense of algebraic geometry).

identities equations formulated using operations structure allows, , variables tacitly universally quantified on relevant universe. identities contain no connectives, existentially quantified variables, or relations of kind other allowed operations. study of varieties important part of universal algebra. algebraic structure in variety may understood quotient algebra of term algebra (also called absolutely free algebra ) divided equivalence relations generated set of identities. so, collection of functions given signatures generate free algebra, term algebra t. given set of equational identities (the axioms), 1 may consider symmetric, transitive closure e. quotient algebra t/e algebraic structure or variety. thus, example, groups have signature containing 2 operators: multiplication operator m, taking 2 arguments, , inverse operator i, taking 1 argument, , identity element e, constant, may considered operator takes 0 arguments. given (countable) set of variables x, y, z, etc. term algebra collection of possible terms involving m, i, e , variables; example, m(i(x), m(x,m(y,e))) element of term algebra. 1 of axioms defining group identity m(x, i(x)) = e; m(x,e) = x. axioms can represented trees. these equations induce equivalence classes on free algebra; quotient algebra has algebraic structure of group.

several non-variety structures fail varieties, because either:

structures axioms unavoidably include nonidentities among important ones in mathematics, e.g., fields , division rings. although structures nonidentities retain undoubted algebraic flavor, suffer defects varieties not have. example, product of 2 fields not field.