Word Hyperbolic Semigroups
  The study of word hyperbolic groups is a prominent topic in geometric group theory; however word hyperbolic groups are defined by a geometric condition which does not extend naturally to semigroups. We propose a linguistic definition. Roughly speaking a semigroup is word hyperbolic if its multiplication table is a context free language. For groups this definition is equivalent to the original geometric one. We also briefly consider word problems of semigroups. 
Andrew Duncan and Robert H. Gilman