1 Introduction In order to illustrate the main idea of the paper let us consider the following easy example. Let E be an alphabet. Take a horizontal straight line R. Take an interval on R and divide it into a union of subintervals. Fix one of two possible orientations in each of these subintervals and label each of them by a letter from E. As a result we get an oriented labelled graph. Let us call such graphs linear diagrams over E. If A is a linear diagram then let us denote the leftmost vertex of A by i(A) and the rightmost vertex of A by r(A). Notice that these two vertices form the boundary of A. a b a b a b *•-• ++ *- Figure 1. We also consider trivial linear diagrams A with i(A) = r(A). Let us not distinguish two linear diagrams which can be transformed one into another by an isotopy of the line which preserves labels and orientation. For example all trivial linear diagrams are considered equal. Given two linear diagrams Ai and A2 one can concatenate them by identifying T ( A I ) with the initial vertex of 1(^2). Thus the set of all linear diagrams forms a monoid with the trivial diagram playing the role of a unit. This monoid is obviously free. Now let us define an equivalence relation on the set of linear diagrams over E. A 2-edge subgraph of a linear diagram is called a dipole if it has one _ f the following forms: a a a a ++** •-* •-• Figure 2. If a linear diagram A contains such a dipole then we can remove these two edges and the middle vertex from A and identify the remaining two vertices. We call this operation a reduction of a dipole. The opposite operation is called an insertion of a dipole. Let us call two linear diagrams equivalent if one of them can be obtained from another one by a series of these operations. It is clear that this equivalence relation is a congruence on the semigroup of linear diagrams and that the factor-semigroup is a group (the trivial diagram plays the role of an identity element). This group is obviously isomorphic to the free group on the set E. In this paper, we study 2-dimensional analogies of this idea: semigroup diagrams, monoid pictures, annular diagrams, cylindric pictures and braided pictures. While the groups of linear diagrams are all free, we get a large class of groups which are rep- resentable by 2-dimensional semigroup diagrams. Semigroup diagrams, are well-known geometrical objects used in the study of Thue systems (=semigroup presentations). They were first formally introduced by Kashintsev [16], see also Remmers [29], Stallings [34] 1
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