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1. Introduction THE use of groups with operators to generalize certain basic theorems of group theory is familiar, and has proved fruitful as a method of unification. However, this method is somewhat unsatisfactory in that it fails to deal adequately with the elementary theory of rings. A ring R can be considered as a group with operators in which the group concerned is the additive group of R and the operators are the left and right multiplications by elements of R; but then the operator homomorphisms of R are not necessarily ring homomorphisms, and we cannot deduce the homomorphism theorems or the Jordan-Holder theorems for rings from the corresponding theorems for groups with operators. Further, the admissible subgroups in this case are the ideals of R, and any discussion of the subrings or the composition series of R is, to say the least, made difficult. The fact is that we are trying to fit a square peg into a round hole; for a ring is most naturally viewed, not as a group admitting a set of unary operators, but as a group admitting a single binary operator.We propose therefore to examine a more general situation in which a group G admits a set Q, of operators, these operators being unary, binary, or of any finite weight. Thus a linear algebra is a group admitting a binary operator (multiplication) and a set of unary operators (the base field), and fits naturally into this scheme. So also do Lie and Jordan triple systems,f in which a ternary operator is involved. The only restriction we impose on the operators is the requirement that the zeroj element of the group should form a one-element subalgebra. Ordinary groups with operators appear as the special case in which all the operators are unary, and it should be noticed that we do not require these unary operators to be endomorphisms of the group but merely mappings of the group into itself which leave the zero element unchanged.In this setting we establish a great variety of theorems which can be immediately applied to the special cases mentioned above by choosing the operators appropriately. The homomorphism theorems and the Jordan-Holder theorems are proved in their simplest form and are not subject to

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The title of this paper is chosen to imitate that of the paper by van Kampen [10] which gave some basic computational rules for the fundamental group TT X { Y, £) of a based space (an earlier more special result is due to Seifert [14]).In [1] results more general than van Kampen's were obtained in terms of fundamental groupoids. The advantage of the use of groupoids is that one obtains an easy description of the fundamental groupoid of a union of spaces even when the spaces and their intersections are not pathconnected ; in such cases, the computation of the fundamental group is greatly simplified by using groupoids.To obtain analogous results in dimension 2 we make essential use of a kind of double groupoid first described in [4]. A major aim is to introduce the homotopy double groupoid p(X, Y,Z) defined for any triple (X, Y,Z) of spaces such that every loop in Z is contractible in Y. The methods of [1] are generalized to give results on p (X, Y,Z). We obtain, as algebraic consequences, results on the second relative homotopy group 7T 2 {X, Y, t) in the form of computational rules for the crossed moduleWe are grateful to referees for helpful comments.