Asymmetric regular types
arXiv:1312.0222 · doi:10.1016/j.apal.2014.09.003
Abstract
We study asymmetric regular types. If is regular and -asymmetric then there exists a strict order such that Morley sequences in over are strictly increasing (we allow Morley sequences to be indexed by elements of a linear order). We prove that for all maximal Morley sequences in over consisting of elements of have the same (linear) order type, denoted by $\Inv_{\frak p,A}(M)$, which does not depend on the particular choice of the order witnessing the asymmetric regularity. In the countable case we determine all possibilities for $\Inv_{\frak p,A}(M)$: either it can be any countable linear order, or in any it is a dense linear order (provided that it has at least two elements). Then we study relationship between $\Inv_{\frak p,A}(M)$ and $\Inv_{\frak q,A}(M)$ when and are strongly regular, -asymmetric, and such that $\frak p_{\strok A}$ and $\frak q_{\strok A}$ are not weakly orthogonal. We distinguish two kinds on non-orthogonality: bounded and unbounded. In the bounded case we prove that $\Inv_{\frak p,A}(M)$ and $\Inv_{\frak q,A}(M)$ are either isomorphic or anti-isomorphic. In the unbounded case, $\Inv_{\frak p,A}(M)$ and $\Inv_{\frak q,A}(M)$ may have distinct cardinalities but we prove that their Dedekind completions are either isomorphic or anti-isomorphic. We provide examples of all four situations.