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- W99230893 abstract "It has been proved by Lachlan [3] and Lerman [4], respectively, that all countable distributive lattices (with least element) and all finite lattices are embeddable as initial segments of the upper semilattice ~ of degrees of recursive unsolvability; these are the two most significant steps so far taken towards an understanding of the general theory of initial segments of ~. Both of these theorems extend the special case of finite distributive lattices, which is particularly interesting because, as Thomason [9] has recently observed, it implies the undecidability of the elementary theory of ~ (Lachlan had previously observed that this followed from his theorem). Lachlan's theorem also generalised a particularly useful special case previously obtained by Hugill [2]: every countable linear ordering (with least element) is embeddable as an initial segment of 9. The most striking consequence of this is the refutation of the strong homogeneity-conjecture; this was first noticed by Feiner [I] and we shall indicate how to considerably strengthen his observation in ~5 of this paper. The homogeneityconjecture, which remains open, asserts the existence, for any degree g, of an isomorphism between (~ ,~) and (~ ~),~); the stronger conjecture asserted the existence of an isomorphism between (~ ,~,J) and (~ g),~,J), where J is the jump operator." @default.
- W99230893 created "2016-06-24" @default.
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- W99230893 date "1972-01-01" @default.
- W99230893 modified "2023-09-25" @default.
- W99230893 title "Initial segments and implications for the structure of degrees" @default.
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- W99230893 doi "https://doi.org/10.1007/bfb0059550" @default.
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