Portal de Periódicos da CAPES

Stable finitely homogeneous structures

1986; American Mathematical Society; Volume: 296; Issue: 2 Linguagem: Inglês

10.1090/s0002-9947-1986-0846608-8

1088-6850

Gregory Cherlin, A. H. Lachlan,

Algebraic structures and combinatorial models

Let L L be a finite relational language and Hom ⁡ ( L , ω ) \operatorname {Hom}(L,\omega ) denote the class of countable L L -structures which are stable and homogeneous. The main result of the paper is that there exists a natural number c ( L ) c(L) such that for any transitive M ∈ Hom ⁡ ( L ; ω ) \mathcal {M} \in \operatorname {Hom}(L;\omega ) , if E E is a maximal 0 0 -definable equivalence relation on M \mathcal {M} , then either | M / E | > c ( L ) |\mathcal {M}/E| > c(L) , or M / E \mathcal {M}/E is coordinatizable. In an earlier paper the second author analyzed certain subclasses Hom ⁡ ( L , r ) ( r > ω ) \operatorname {Hom}(L, r)\ (r > \omega ) of Hom ⁡ ( L , ω ) \operatorname {Hom}(L,\omega ) for all sufficiently small r r . Thus the earlier analysis now applies to Hom ⁡ ( L , ω ) \operatorname {Hom}(L,\omega ) .