A connected graph G is said to be z-homogeneous if any isomorphism between finite connected induced subgraphs of G extends to an automorphism of G. Finite z-homogeneous graphs were classified in [17]. We show that z-homogeneity is... more
A connected graph G is said to be z-homogeneous if any isomorphism between finite connected induced subgraphs of G extends to an automorphism of G. Finite z-homogeneous graphs were classified in [17]. We show that z-homogeneity is equivalent to finite-transitivity on the class of infinite locally finite graphs. Moreover, we classify the graphs satisfying these properties. Our study of bipartite
Using the Lascar inequalities, we show that any finite rank $\delta$-closed subset of a quasiprojective variety is definably isomorphic to an affine $\delta$-closed set. Moreover, we show that if X is a finite rank subset of the... more
Using the Lascar inequalities, we show that any finite rank $\delta$-closed subset of a quasiprojective variety is definably isomorphic to an affine $\delta$-closed set. Moreover, we show that if X is a finite rank subset of the projective space $\mathbb{P}^n$ and a is a generic point of $\mathbb{P}^n$, then the projection from a is injective on X. Finally we prove that if RM = RC in DCF$_0$, then RM = RU.
The Journal of Symbolic Logic Volume 67, Number 3, Sept. 2002 ON LASCAR RANK AND MORLEY RANK OF DEFINABLE GROUPS IN DIFFERENTIALLY CLOSED FIELDS ANAND PILLAY AND WAI YAN PONG Abstract. Morley rank and Lascar rank are equal on generic... more
The Journal of Symbolic Logic Volume 67, Number 3, Sept. 2002 ON LASCAR RANK AND MORLEY RANK OF DEFINABLE GROUPS IN DIFFERENTIALLY CLOSED FIELDS ANAND PILLAY AND WAI YAN PONG Abstract. Morley rank and Lascar rank are equal on generic types of ...
The Journal of Symbolic Logic Volume 67, Number 3, Sept. 2002 ON LASCAR RANK AND MORLEY RANK OF DEFINABLE GROUPS IN DIFFERENTIALLY CLOSED FIELDS ANAND PILLAY AND WAI YAN PONG Abstract. Morley rank and Lascar rank are equal on generic... more
The Journal of Symbolic Logic Volume 67, Number 3, Sept. 2002 ON LASCAR RANK AND MORLEY RANK OF DEFINABLE GROUPS IN DIFFERENTIALLY CLOSED FIELDS ANAND PILLAY AND WAI YAN PONG Abstract. Morley rank and Lascar rank are equal on generic types of ...
Let X be a δ-variety over some δ-field K. Denote by tdδ(X/K), or simply tdδ(X) if the ground field is understood, the δ-transcendental degree of K〈X〉 over K. Suppose tdδ(X)=d; Johnson [Comment. Math. Helv.44 (1969), 207–216] showed that... more
Let X be a δ-variety over some δ-field K. Denote by tdδ(X/K), or simply tdδ(X) if the ground field is understood, the δ-transcendental degree of K〈X〉 over K. Suppose tdδ(X)=d; Johnson [Comment. Math. Helv.44 (1969), 207–216] showed that there is an increasing chain of δ-subvarieties of length ωd in X. The question, also known as the Kolchin Catenary Problem, is: Given a point x∈X, is there an increasing chain of δ-subvarieties of length ωd starting at x? We will give an affirmative answer to this question if X is an algebraic variety.
A connected graph G is said to be z-homogeneous if any isomorphism between finite connected induced subgraphs of G extends to an automorphism of G. Finite z-homogeneous graphs were classified in [17]. We show that z-homogeneity is... more
A connected graph G is said to be z-homogeneous if any isomorphism between finite connected induced subgraphs of G extends to an automorphism of G. Finite z-homogeneous graphs were classified in [17]. We show that z-homogeneity is equivalent to finite-transitivity on the class of infinite locally finite graphs. Moreover, we classify the graphs satisfying these properties. Our study of bipartite
