Semigroups Characterizing Hypergraphs

Various semigroups can be associated with graphs and hypergraphs. Mainly, these have been endomorphism monoids. However, other types of semigroups (e.g. restrictive semigroups of partial extensive transformations of graphs) have been considered. A good survey of semigroups connected with graphs is given in [5]. In this paper, a new type of semigroups connected with hypergraphs is considered. We call such semigroups 'hypergraph semigroups'. It turns out that every hypergraph is completely characterized by its hypergraph semigroup (in the sense that two hypergraph semigroups are isomorphic if and only if their hypergraphs are isomorphic). We also give an algebraic characterization of hypergraph semigroups. A special class of hypergraph semigroups (those without nontrivial one-sided annihilators) was considered earlier by the author [6J (who called them 'matrix O-bands') and in [3J (where they were called 'rectangular O-bands'). Various properties of such semigroups are given in these papers. Translational hulls of reductive hypergraph semigroups are described in [4]. A hypergraph is a triple 'Je = (V, E, I), where V and E are sets and I c V x E is a binary relation between elements of V and E. The inclusion sign c is reflexive, i.e. the case I = V x E is possible. Elements of V are called vertices, elements of E are called edges, (v, e)E I is interpreted as 'v and e are incident', or 'v belongs to e', or 'e contains v', and I is called the incidence relation of 'Je. A vertex (edge) is called isolated if it does not belong to (does not contain) any edge (any vertex). Two different edges may have exactly the same vertices (therefore our hypergraphs are something like 'multi-hypergraphs'), and two different vertices may belong to exactly the same edges. If 1= 0, 'Je is called empty. This definition of a hypergraph is probably the most general (cf. [IJ, [7]). Obviously, ordinary (non-directed) graphs and multi-graphs are special cases of hypergraphs (as are various geometries and combinatorial designs). With each hypergraph 'Je we associate a groupoid G('Je). The set of elements of G('Je) is V x E. Multiplication in G( 'Je) is partially defined: a product (VI> ej)( V2, e2) exists if and only if (V2, e.) E I, in which case (v h e\)( V 2, e2)= (v h e2)' Clearly, multiplication in G('Je) is everywhere defined if and only if I = V x E. To make G('Je) into a groupoid with everywhere defined multiplication we perform the usual trick of extension: Let 0 be an element not in G( 'Je). Define S( 'Je) = G( 'Je) u O. Define multiplication in S( 'Je) as follows: all products existing in G( 'Je) are preserved, all other products are defined to be O. Thus 0 is the zero of S( 'Je), and (v h e\)( V2, e2 ) = 0 whenever e j does not contain V2'