On Intervals in Relational Structures

Throughout this paper we shall use the letters R and A to denote a finitary relation and its underlying universe. Formally, an n-ary relation (n positive integer) on A is defined as a mapping R : All --f (0, I}. For any B A , the restriction RIB is defined by (RIB) (q, . . ., zll) = R(z,, . . ., z,,) for every (zl, . . ., z,J E Bn. An isomorphism between two restrictions of R is called a local autornorphisrn of R. In the fifties FRAISSB defined an interval to be a subset I of A such that every local automorphism of RII, extended by the identity on A \ I , gives a local automorphism of R (see [lo, 111). Here we shall study EL more restrictive concept, introduced in [9] under the name of “part>ie solidaire”: a subset S E A is called a strong interval of R if for every (zl, . . ., zll) E A“ \ PI, and for every 1 5 i 5 n such that zi E S , the value of R(x1, . . . , zlj ) is unaltered by changing zi into any other element of S. It is easily verified t,hat all strong intervals are int.ervals.