Nowadays algebra is understood basically as the general theory of
algebraic oper- ations and relations. It is characterised by a
considerable intrinsic naturalness of its initial notions and problems,
the unity of its methods, and a breadth that far exceeds that of its
basic concepts. It is more often that its power begins to be displayed
when one moves outside its own limits. This characteristic ability is
seen when one investigates not only complete operations, but partial
operations. To a considerable extent these are related to algebraic
operators and algebraic operations. The tendency to ever greater
generality is amongst the reasons that playa role in explaining this
development. But other important reasons play an even greater role.
Within this same theory of total operations (that is, operations defined
everywhere), there persistently arises in its different sections a
necessity of examining the emergent feature of various partial
operations. It is particularly important that this has been found in
those parts of algebra it brings together and other areas of mathematics
it interacts with as well as where algebra finds applica- tion at the
very limits of mathematics. In this connection we mention the theory of
the composition of mappings, category theory, the theory of formal
languages and the related theory of mathematical linguistics, coding
theory, information theory, and algebraic automata theory. In all these
areas (as well as in others) from time to time there arises the need to
consider one or another partial operation.
