Injective compact distributive lattices

Choe, Tae Ho

doi:10.1090/S0002-9939-1973-0309818-0

Injective compact distributive lattices
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by Tae Ho Choe

Proc. Amer. Math. Soc. 37 (1973), 241-245

DOI: https://doi.org/10.1090/S0002-9939-1973-0309818-0

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Abstract:

In the category of zero-dimensional compact distributive lattices and their continuous lattice-homomorphisms, it is firstly shown that the injective objects are precisely the compact Boolean algebras, and each object has an injective envelope. Secondly, it is shown that the category of compact distributive lattices and their continuous lattice-homomorphisms has no non-trivial injective object.

References

Lee W. Anderson, One dimensional topological lattices, Proc. Amer. Math. Soc. 10 (1959), 715–720. MR 107678, DOI 10.1090/S0002-9939-1959-0107678-7
B. Banaschewski and G. Bruns, Injective hulls in the category of distributive lattices, J. Reine Angew. Math. 232 (1968), 102–109. MR 237607
B. Banaschewski and G. Bruns, Categorical characterization of the MacNeille completion, Arch. Math. (Basel) 18 (1967), 369–377. MR 221984, DOI 10.1007/BF01898828
Raymond Balbes, Projective and injective distributive lattices, Pacific J. Math. 21 (1967), 405–420. MR 211927
Tae Ho Choe, On compact topological lattices of finite dimension, Trans. Amer. Math. Soc. 140 (1969), 223–237. MR 240012, DOI 10.1090/S0002-9947-1969-0240012-1
Tae Ho Choe, Locally compact lattices with small lattices, Michigan Math. J. 18 (1971), 81–85. MR 282343
E. B. Davies, The existence of characters on topological lattices, J. London Math. Soc. 43 (1968), 217–220. MR 224513, DOI 10.1112/jlms/s1-43.1.217
B. Eckmann and A. Schopf, Über injektive Moduln, Arch. Math. (Basel) 4 (1953), 75–78 (German). MR 55978, DOI 10.1007/BF01899665
Paul R. Halmos, Injective and projective Boolean algebras, Proc. Sympos. Pure Math., Vol. II, Amer. Math. Soc., Providence, RI, 1961, pp. 114–122. MR 137671
Barry Mitchell, Theory of categories, Pure and Applied Mathematics, Vol. XVII, Academic Press, New York-London, 1965. MR 202787
Katsumi Numakura, Theorems on compact totally disconnected semigroups and lattices, Proc. Amer. Math. Soc. 8 (1957), 623–626. MR 87032, DOI 10.1090/S0002-9939-1957-0087032-5