REPRESENTATION THEORY AND RELATED TOPICS SEMINAR

ABSTRACT: The notion of an ideal of an algebra from a variety of universal algebras was introduced by A. Ursini. It generalizes the notions of a normal divisor of a group, a normal subloop of a loop, an ideal of a ring, an ideal of a Boolean algebra, etc. An ideal of an algebra from a variety is defined as a non-empty subset which is closed under all so-called ideal terms for arbitrary fixed values of certain variables. It is well-known that for many familiar varieties (groups, loops, rings, Boolean algebras, etc.) there is a finite set T of terms such that the set of all ideal terms in this definition can be replaced by the set T. In 1984 A. Ursini proved that this is not a mere coincidence, and such a finite set T exists for an arbitrary finitary ideal-determined variety. However the proof did not give the explicit form of terms of that set. In the present work we give the terms from a set T explicitly, for the case of an arbitrary finitary classically ideal-determined variety. In fact, we give several finite sets of this kind. For the variety of groups (resp. rings) one of the found ideal term sets contains the typical term (resp. terms) from the well-known definition of a normal divisor of a group (resp. an ideal of a ring).