Geodesic
Numbered Distributive Semilattices
Matematičeskie trudy, Tome 9 (2006) no. 2, pp. 109-132.

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In this article, we consider several definitions of a Lachlan semilattice; i. e., a semilattice isomorphic to a principal ideal of the semilattice of computably enumerable $m$-degrees. We also answer a series of questions on constructive posets and prove that each distributive semilattice with top and bottom is a Lachlan semilattice if it admits a $\Sigma^0_3$-representation as an algebra but need not be a Lachlan semilattice if it admits a $\Sigma^0_3$-representation as a poset. The examples are constructed of distributive lattices that are constructivizable as posets but not constructivizable as join (meet) semilattices. We also prove that every locally lattice poset (in particular, every lattice and every distributive semilattice) possessing a $\Delta^0_2$-representation is positive. 
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S. Yu. Podzorov. Numbered Distributive Semilattices. Matematičeskie trudy, Tome 9 (2006) no. 2, pp. 109-132. http://geodesic.mathdoc.fr/item/MT_2006_9_2_a4/

[1] Goncharov S. S., Schetnye bulevy algebry i razreshimost, Nauchnaya kniga, Novosibirsk, 1996 | MR

[2] Grettser G., Obschaya teoriya reshetok, Mir, M., 1982 | MR

[3] Denisov S. D., “Stroenie verkhnei polu reshetki rekursivno perechislimykh $m$-stepenei i smezhnye voprosy. I”, Algebra i logika, 17:6 (1978), 643–683 | MR | Zbl

[4] Ershov Yu. L., Teoriya numeratsii, Nauka, M., 1977 | MR

[5] Ershov Yu. L., “Polureshetki Rodzhersa konechnykh chastichno uporyadochennykh mnozhestv”, Algebra i logika, 45:1 (2006), 44–84 | MR | Zbl

[6] Podzorov S. Yu., “O lokalnom stroenii polu reshetok Rodzhersa $\Sigma_n^0$-vychislimykh numeratsii”, Algebra i logika, 44:2 (2005), 148–172 | MR | Zbl

[7] Podzorov S. Yu., “Ob opredelenii lakhlanovskoi polureshetki”, Sib. mat. zhurn., 47:2 (2006), 383–393 | MR | Zbl

[8] Rodzhers X., Teoriya rekursivnykh funktsii i effektivnaya vychislimost, Mir, M., 1972 | MR

[9] Lachlan A. H., “Recursively enumerable many-one degrees”, Algebra i Logika, 11:3 (1972), 326–358 | MR | Zbl