Abstract
A study is made of Boolean product representations of bounded lattices over the Stone space of their centres. Special emphasis is placed on relating topological properties such as clopen or regular open equalizers to their equivalent lattice theoretic counterparts. Results are also presented connecting various properties of a lattice with properties of its individual stalks.
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References
Balbes, R. and Dwinger, P. (1974) Distributive Lattices, University of Missouri Press.
Blyth, T. S. and Janowitz, M. F. (1972) Residuation Theory, Pergamon Press.
Banaschewski, B. (1956) Hüllensysteme und Erweiterungen von Quasi-Ordnung, Z. Math. Logik Grundl. Math. 2, 35–46.
Birkhoff, G. (1944) Subdirect unions in universal algebras, Bull. Amer. Math. Soc. 50, 764–768.
Burgess, W. D. and Hoffman, K. H. (1966) Pierce sheaves of noncommutative rings, Comm. in Algebra 4, 51–75.
Burris, S. and Sankappanavar, H. P. (1981) A Course in Universal Algebra, Springer-Verlag, New York.
Burris, S. and Werner, H. (1979) Sheaf constructions and their elementary properties, Trans. Amer: Math. Soc. 248, 269–309.
Carson, A. (1989) Model completions, ring representations and the topology of the Pierce sheaf, Pitman research notes in mathematics series 209.
Carson, A., personal communication.
Cignoli, R. (1972) Representation of Lukasiewicz and Post algebras by continuous functions, Colloq. Math. 24, 127–138.
Cignoli, R. (1978) The lattice of global sections of sheaves of chains over Boolean spaces, Algebra Universalis 8, 357–373.
Comer, S. D. (1971) Representations by algebras of sections over Boolean spaces, Pacific J. Math. 38, 29–38.
Comer, S. D. (1978) Elementary properties of structures of sections, Bol. Soc. Mat. Mexicana Ser. 2, 78–85.
Dauns, J. and Hofmann, K. H. (1966) The representation of biregular rings by sheaves, Math. Zeit. 91, 103–123.
Davey, B. A. (1973) Sheaf spaces and sheaves of universal algebras, Math. Zeit. 134, 275–290.
Dilworth, R. P. (1945) Lattices with unique complements, Trans. Amer. Math. Soc. 57, 123–154.
Epstein, G. (1960) The lattice theory of Post algebras, Trans. Amer. Math. Soc. 95, 300–317.
Epstein, G. and Horn, A. (1974) P-algebras, an abstraction from Post algebras, Algebra Universalis 4, 195–206.
Epstein, G. and Horn, A. (1976) Logics which are characterized by subresiduated lattices, Zeitsch. f. Math. Logik und Grundlagen der Mathematik 22, 199–210.
Harding, J. (1993) Completions of orthomodular lattices II, Order 10, 283–294.
Harding, J. and Janowitz, M. F. A bundle representation for continuous geometries, Advances in Applied Math. To appear.
Janowitz, M. F. (1973) On a paper by Iqbalunisa, Fund. Math. 78, 177–182.
Janowitz, M. F. (1978) Note on the center of a lattice, Math. Slovaca 28, 235–242.
Kopperman, R. (1972) Model Theory and Its Applications, Allyn and Bacon, Boston.
Macintyre, A. (1973) Model-completeness for sheaves of structures, Fund. Math. 81, 73–89.
MacNeille, H. M. (1937) Partially ordered sets, Trans. Amer. Math. Soc. 42, 416–460.
Maeda, F. (1959) Decomposition of general lattices into direct summands of types I, II and III, J. Sci. Hiroshima Univ., Ser. A 23, 151–170.
Maeda, F. and Maeda, S. (1970) Theory of Symmetric Lattices, Springer-Verlag, New York.
Pierce, R. S. (1967) Modules over commutative regular rings, Memoirs of the AMS, 70.
Stone, M. H. (1936) The theory of representations for Boolean algebras, Trans. Amer. Math. Soc. 40, 37–111.
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Communicated by H. A. Priestley
Research supported by the Natural Sciences and Engineering Research Council of Canada.
Research supported by ONR Grant N00014-90-J-1008.
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Crown, G.D., Harding, J. & Janowitz, M.F. Boolean products of lattices. Order 13, 175–205 (1996). https://doi.org/10.1007/BF00389840
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DOI: https://doi.org/10.1007/BF00389840