Abstract
Recently there has been significant progress in the study of powers of ordered sets. Much of this work has concerned cancellation laws for powers and uses these two steps. First, logarithmic operators are introduced to transform cancellation problems for powers into questions involving direct product decompositions. Second, refinement theorems for direct product decompositions are brought to bear. Here we present two results with the aim of highlighting these steps.
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References
K.Baclawski and A.Björner (1979) Fixed points in partially ordered sets, Adv. Math. 31, 263–287.
H. Bauer (1981) Garben und Automorphismen geordneter Mengen, PhD Thesis, Darmstadt.
H. Bauer and R. Wille (1983) A proof of Hashimoto's refinement theorem, preprint.
H.Bauer, K.Keimel, and R.Kohler (1981) Verfeinerungs — und Kurzungssatze für Produkte geordneter topologischer Raume und für Funktionen (-Halb-) Verbande, Lecture Notes in Math. 871, 20–44.
C.Bergman, R.McKenzie, and Sz.Nagy (1982) How to cancel a linearly ordered exponent, Colloq. Math. Soc. Janos Bolyai 21, 87–94.
G.Birkhoff (1937a) Extended arithmetic, Duke Math. J. 3, 311–316.
G.Birkhoff (1937b) Rings of sets, Duke Math. J. 3, 443–454.
G.Birkhoff (1940) Lattice Theory, 1st edn., American Mathematical Society, Providence, Rhode Island.
G.Birkhoff (1942) Generalized arithmetic, Duke Math. J. 9, 283–302.
G.Birkhoff (1967) Lattice Theory, 3rd edn., American Mathematical Society, Providence, Rhode Island.
C. C.Chang, B.Jónsson, and A.Tarski (1964) Refinement properties for relational structures, Fund. Math. 55, 249–281.
B. A.Davey and D.Duffus (1982) Exponentiation and duality, in Ordered Sets (ed. I.Rival), D. Reidel, Dordrecht, pp. 43–95.
B. A.Davey, D.Duffus, R. W.Quakenbush, and I.Rival, Exponents of finite simple lattices (1978) J. London Math. Soc. (2) 17, 203–211.
B. A.Davey and I.Rival (1982) Exponents of lattice-ordered algebras, Alg. Univ. 14, 87–98.
D. Duffus (1984) Automorphisms and products of ordered sets, Alg. Univ. (to appear).
D.Duffus, B.Jónsson, and I.Rival (1978) Structure results for function lattices, Canad. J. Math. 30, 392–400.
D.Duffus and I.Rival (1978) A logarithmic property for exponents of partially ordered sets, Canad. J. Math. 30, 797–807.
D.Duffus and I.Rival (1981) A structure theory for ordered sets, Discrete Math. 35, 53–118.
D.Duffus and R.Wille (1979) A theorem on partially ordered sets of order preserving mappings, Proc. Amer. Math. Soc. 76, 14–16.
D.Duffus and R.Wille (1982) Automorphism groups of function lattices, Colloq. Math. Soc. Janos Bolyai 29, 203–207.
E.Fuchs (1965) Isomorphismus der Kardinalpotenzen, Arch. Math (Brno) 1, 83–93.
J.Hashimoto (1948) On the product decomposition of partially ordered sets, Math. Japon. 1, 120–123.
J.Hashimoto (1951) On direct product decomposition of partially ordered sets, Ann. Math. 54, 315–318.
B.Jónsson (1982a) Arithmetic of ordered sets, in Ordered Sets (ed. I.Rival), D. Reidel, Holland, pp. 3–41.
B.Jónsson (1982b) Powers of partially ordered sets: the automorphism group, Math. Scand. 51 121–141.
B.Jónsson and R.McKenzie (1982) Powers of partially ordered sets: cancellation and refinement properties, Math. Scand. 51, 87–120.
L.Lovász (1967) Operations with structures, Acta Math. Acad. Sci. Hungar. 18, 321–328.
R.McKenzie (1971) Cardinal multiplication of structures with a reflexive relation, Fund. Math. 70, 59–101.
R.McKenzie and R.Quackenbush (1981) The spectrum of a lattice-primal algebra, Discrete Math. 35, 157–164.
M. Novotny (1960) Uber gewisse Eigenschaften von Kardinaloperationen, Spisy Prirod Fac. Univ. Brno, pp. 465–484.
H. A.Priestley (1970) Representation of distributive lattices by means of ordered Stone spaces, Bull. London Math. Soc. 2, 186–190.
H. A.Priestley (1972) Ordered topological spaces and the representation of distributive lattices, Proc. London Math. Soc. (3) 24, 507–530.
R.Quackenbush (1982) Enumeration in classes of ordered structures, in Ordered Sets (ed. I.Rival), D. Reidel, Dordrecht, pp. 523–554.
I.Rival (ed.) (1982) Ordered Sets, D. Reidel, Dordrecht.
G.Sabidussi (1960) Graph multiplication, Math. Z. 72, 446–457.
R.Wille (1980) Cancellation and refinement results for function lattices, Houston J. Math. 6, 431–437.
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Communicated by B. Jōnsson
Supported by NSF grant MCS 83-02054
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Duffus, D. Powers of ordered sets. Order 1, 83–92 (1984). https://doi.org/10.1007/BF00396275
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DOI: https://doi.org/10.1007/BF00396275