The FraserHorn and Apple properties
Authors:
Joel Berman and W. J. Blok
Journal:
Trans. Amer. Math. Soc. 302 (1987), 427465
MSC:
Primary 08B20; Secondary 03G25, 08A40
MathSciNet review:
891629
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Abstract: We consider varieties in which finite direct products are skewfree and in which the congruence lattices of finite directly indecomposables have a unique coatom. We associate with a family of derived varieties, : a variety in is generated by algebras where the universe of consists of a congruence class of the coatomic congruence of a finite directly indecomposable algebra and the operations of are those of that preserve this congruence class. We also consider the prime variety of , denoted , generated by all finite simple algebras in . We show how the structure of finite algebras in is determined to a considerable extent by and . In particular, the free algebra on generators, , has as many directly indecomposable factors as and the structure of these factors is determined by the varieties . This allows us to produce in many cases explicit formulas for the cardinality of . Our work generalizes the structure theory of discriminator varieties and, more generally, that of arithmetical semisimple varieties. The paper contains many examples of algebraic systems that have been investigated in different contexts; we show how these all fit into a general scheme.
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 [W]
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 [W]
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 [B]
 Bosbach [1969], Komplementare Halbgruppen, Fund. Math. 64, 257287. MR 0260902 (41:5522)
 [S]
 Burris [1982], Discriminator polynomials and arithmetical varieties, manuscript.
 [S]
 Burris and J. Lawrence [1981], A correction to "Definable principal congruences in varieties of groups and rings," Algebra Universalis 13, 264267. MR 631561 (82j:08008)
 [S]
 Burris and H. Sankappanavar [1981], A course in universal algebra, Graduate Texts in Math., No. 78, SpringerVerlag. MR 648287 (83k:08001)
 [W]
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 [B]
 A. Davey [1977], Weak injectivity and congruence extension in congruencedistributive equational classes, Canad. J. Math. 29, 449459. MR 0441823 (56:217)
 [J]
 Demetrovics, L. Hannak and L. Ronyai [1982], On the free spectra of maximal clones, C. R. Math. Rep. Acad. Sci. Canada 4, 363366. MR 681194 (83m:08012)
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 [A]
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 [G]
 A. Fraser and A. Horn [1970], Congruence relations in direct products, Proc. Amer. Math. Soc. 26, 390394. MR 0265258 (42:169)
 [R]
 S. Freese and J. B. Nation [1973], Congruence lattices of semilattices, Pacific J. Math. 49, 5158. MR 0332590 (48:10916)
 [G]
 Gratzer [1979], Universal algebra, 2nd ed., SpringerVerlag. MR 538623 (80g:08001)
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 Gratzer and E. T. Schmidt [1957], On a problem of M. H. Stone, Acta Math. Acad. Sci. Hungar. 8, 455460. MR 0092763 (19:1154h)
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 [T]
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 [B]
 Jónsson [1967], Algebras whose congruence lattices are distributive, Math. Scand. 21, 110121. MR 0237402 (38:5689)
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 Kàhler [1973], Freie endlich erzeugte Heyting Algebren, Diplomarbeit, Justus Liebig Universität, Giessen.
 [H]
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 [S]
 Mac Lane [1971], Categories for the working mathematician, SpringerVerlag. MR 0354798 (50:7275)
 [R]
 McKenzie [1982], Narrowness implies uniformity, Algebra Universalis 15, 6785. MR 663953 (83i:08003)
 1.
 [1984], A new product of algebras and a type reduction theorem, Algebra Universalis 18, 2969. MR 743456 (86h:08011)
 [R]
 McKenzie and D. Hobby [1986], The structure of finite algebras (tame congruence theory), manuscript.
 [A]
 F. Pixley [1971], The ternary discriminator function in universal algebra, Math. Ann. 191, 167180. MR 0292738 (45:1820)
 2.
 [1985], Principal congruence formulas in arithmetical varieties, Universal Algebra and Lattice Theory, Lecture Notes in Math., vol. 1149, Springer, pp. 238254. MR 823019 (87e:08008)
 [J]
 Płonka [1971], On free algebras and algebraic decomposition of algebras from some equational classes defined by regular equations, Algebra Universalis 1, 261264. MR 0294221 (45:3294)
 [E]
 L. Post [1921], Introduction to a general theory of elementary propositions, Amer. J. Math. 43, 163185. MR 1506440
 [R]
 W. Quackenbush [1974], Structure theory for equational classes generated by quasiprimal algebras, Trans. Amer. Math. Soc. 187, 127145. MR 0327619 (48:5961)
 [M]
 F. Raca [1969], The class of functions of the threevalued logic that corresponds to the first matrix of Jas'kovski, Problemy Kibernet. 21, 185214. (Russian) MR 0307880 (46:6995)
 [W]
 Taylor [1975], The fine spectrum of a variety, Algebra Universalis 5, 263303. MR 0389716 (52:10547)
 [M]
 Tokarz [1980], Essays in matrix semantics of relevant logics, Polish Acad Sci., Institute of Philosophy and Sociology, Warszawa. MR 603277 (82h:03016)
 [H]
 Werner [1970], Eine Charakterisierung funktional vollständiger Algebren, Arch. Math. (Basel) 21, 381385. MR 0269574 (42:4469)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198708916293
PII:
S 00029947(1987)08916293
Keywords:
Free algebra,
FraserHorn Property,
arithmetical variety,
semisimple algebra,
directly indecomposable,
pseudocomplemented lattice,
congruence lattice,
clone of operations
Article copyright:
© Copyright 1987
American Mathematical Society
