In this work we enrich the geometric method of semigroup diagrams to study semigroup presentations. We introduce a process of reduction on semigroup diagrams which leads to a natural way of multiplying semigroup diagrams associated with a given semigroup presentation. With respect to this multiplication the set of reduced semigroup diagrams is a groupoid. The main result is that the groupoid [Formula: see text] of reduced semigroup diagrams over the presentation S = <X:R> may be identified with the fundamental groupoid γ (KS) of a certain 2-dimensional complex KS. Consequently, the vertex groups of the groupoid [Formula: see text] are isomorphic to the fundamental groups of the complex KS. The complex we discovered was first considered in the paper of Craig Squier, published only recently. Steven Pride has also independently defined a 2-dimensional complex isomorphic to KS in relation to his work on low-dimensional homotopy theory for monoids. Some structural information about the fundamental groups of the complex KS are presented. The class of these groups contains all finitely generated free groups and is closed under finite direct and finite free products. Many additional results on the structure of these groups may be found in the paper of Victor Guba and Mark Sapir.

We give a graph-theoretic definition for the number of ends of Cayley digraphs for finitely generated semigroups and monoids. For semigroups and monoids, left Cayley digraphs can be very different from right Cayley digraphs. In either case, the number of ends for the Cayley digraph does not depend upon which finite set of generators is used for the semigroup or monoid. For natural numbers m and n, we exhibit finitely generated monoids for which the left Cayley digraphs have m ends while the right Cayley digraphs have n ends. For direct products and for many other semidirect products of a pair of finitely generated infinite monoids, the right Cayley digraph of the semidirect product has only one end. A finitely generated subsemigroup of a free semigroup has either one end or else has infinitely many ends.2000 Mathematics subject classification: primary 20M99. Keywords and phrases: ends, monoid and semigroup, Cayley digraphs. Ends for graphs and digraphsA digraph is a quadruple = (V , E , ι , τ ) where V = V is a set of vertices, E = E is a set of edges and ι , τ : E → V are functions designating initial and terminal vertices for each edge. A graph is a quintuple = (V , E , ι , τ , inv ) where inv is a function E → E and we require axiomatically, for each e ∈ E, that e = inv (e), that inv (inv (e)) = e, that ι (inv (e)) = τ (e) and that τ (inv (e)) = ι (e). We omit the subscripts on the functions ι and τ whenever context makes these unnecessary and we routinely write e −1 for inv (e).When we imagine some geometric realization of a graph, we regard e and e −1 as occupying the same arc of points, but traversing these arcs in opposite directions. In a geometric realization for a digraph, each edge has an associated direction for traversal. We allow loops and multiple edges in graphs and digraphs.A graph (V ϒ , E ϒ , ι ϒ , τ ϒ , inv ϒ ) is a subgraph of (V , E , ι , τ , inv ) if V ϒ and E ϒ are subsets of V and E , respectively, and the functions ι ϒ , τ ϒ and inv ϒ are the respective restrictions of the functions ι , τ and inv to E ϒ . Subdigraphs of

We define the notion of the partial order of ends of the Cayley graph of a semigroup. We prove that the structure of the ends of a semigroup is invariant under change of finite generating set and at the same time is inherited by subsemigroups and extensions of finite Rees index. We prove an analogue of Hopf's Theorem, stating that an infinite group has 1, 2 or infinitely many ends, for left cancellative semigroups and that the cardinality of the set of ends is invariant in subsemigroups and extension of finite Green index in left cancellative semigroups.

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