Abstract
As the class \(\mathcal {PCSL}\) of pseudocomplemented semilattices is a universal Horn class generated by a single finite structure it has a \(\aleph _0\)-categorical model companion \(\mathcal {PCSL}^*\). As \(\mathcal {PCSL}\) is inductive the models of \(\mathcal {PCSL}^*\) are exactly the existentially closed models of \(\mathcal {PCSL}\). We will construct the unique existentially closed countable model of \(\mathcal {PCSL}\) as a direct limit of algebraically closed pseudocomplemented semilattices.
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Adler, J.: The model companion of the class of pseudocomplemented semilattices is finitely axiomatizable. Algebra Univers (2014). doi:10.1007/s00012-014-0297-9
Balbes, R., Horn, A.: Stone lattices. Duke Math. J. 38, 537–545 (1970)
Burris, S.: Model companions for finitely generated universal Horn classes. J. Symb. Log. 49(1), 68–74 (1984)
Burris, S., Werner, H.: Sheaf constructions and their elementary properties. Trans. Am. Math. Soc. 48, 269–309 (1979)
Frink, O.: Pseudo-complements in semilattices. Duke Math. J. 37, 505–514 (1962)
Jones, G.: Pseudocomplemented semilattices. PhD Thesis, UCLA (1972)
Macintyre, A.: Model completeness. In: Barwise, J. (ed.) Handbook of Mathematical Logic, pp. 139–180. North-Holland, Amsterdam (1977)
Schmid, J.: Algebraically closed p-semilattices. Arch. Math. 45, 501–510 (1985)
Wheeler, W.: Model-companions and definability in existentially complete structures. Isr. J. Math. 25, 305–330 (1976)
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Adler, J. The countable existentially closed pseudocomplemented semilattice. Arch. Math. Logic 56, 397–402 (2017). https://doi.org/10.1007/s00153-017-0527-x
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DOI: https://doi.org/10.1007/s00153-017-0527-x