Injective envelope of graphs and transition systems Original Research Article

To each word a on a two-letter alphabet A:={a, b}, we associate a finite reflexive oriented graph L¬u. These graphs generate the variety ARG of absolute retracts (with respect to isometric embedding) and are indecomposable members of this variety. They turn out to be injective envelopes of particular two-element metric spaces over the Heyting algebra F(A∗) consisting of final segments of A∗ equipped with the Higman ordering. Considering an ordered alphabet A equipped with an involution, we present constructive methods producing the injective envelope of metric spaces over F(A∗). These methods lead to a description of the variety ART of reflexive and involutive transition systems from which the above result is a special case.