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Finite algebras that generate an injectively complete modular variety

Published online by Cambridge University Press:  17 April 2009

Keith A. Kearnes
Affiliation:
Department of Mathematics, Vanderbilt University, Nashville TN 37235, United States of America
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Abstract

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We extend Kollár's result on finitely generated, injectively complete congruence distributive varieties to the congruence modular setting. By doing so we show that, given any finite algebra A of finite type, there is an algorithm to decide whether V(A) is an injectively complete, congruence modular variety.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1991

References

[1]Baldwin, J.T. and Berman, J., ‘The number of subdirectly irreducible algebras in a variety’, Algebra Universalis 5 (1975), 379389.CrossRefGoogle Scholar
[2]Banaschewski, B., ‘Injectivity and essential extensions in equational classes of algebras’, Proceedings of the Conference on Universal Algebra (October 1969), in Queen's Papers in Pure and Applied Mathematics 25, (Kingston, Ontario, 1970), pp. 131147.Google Scholar
[3]Bergman, C. and McKenzie, R., ‘On the relationship of AP, RS and CEP in congruence modular varieties. II’, Proc. Amer. Math. Soc. 103 (1988), 335343.CrossRefGoogle Scholar
[4]Bergman, C. and McKenzie, R., ‘When a minimal variety is a minimal quasivariety’, J. Austal. Math. Soc. (to appear).Google Scholar
[5]Day, A., ‘Injectivity in equational classes of algebras’, Canad. J. Math. 24 (1972), 209220.CrossRefGoogle Scholar
[6]Freese, R. and McKenzie, R., ‘Residually small varieties with modular congruence lattices’, Trans. Amer. Math. Soc. 264 (1981), 419430.CrossRefGoogle Scholar
[7]Freese, R. and McKenzie, R., Commutator theory for congruence modular varieties: LMS Lecture Note Series 125 (Cambridge, 1987).Google Scholar
[8]Kearnes, K.A., ‘On the relationship between AP, RS and CEP’, Proc. Amer. Math. Soc. 4 (1989), 827839.CrossRefGoogle Scholar
[9]Kearnes, K.A., ‘Residual bounds in varieties of modules’, Algebra Universalis (to appear).Google Scholar
[10]Kearnes, K.A., ‘Type restriction in locally finite varieties with the CEP’, Canad. J. Math. (to appear).Google Scholar
[11]Kiss, E., ‘Injectivity and related concepts in modular varieties I-II’, Bull. Austral. Math. Soc. 32 (1985), 3553.Google Scholar
[12]Kollár, J., ‘Injectivity and congruence extension property in congruence distributive equational classes’, Algebra Universalis 10 (1980), 2126.CrossRefGoogle Scholar
[13]McKenzie, R., ‘Congruence extension, Hamiltonian and Abelian properties in locally finite varieties’ (to appear).Google Scholar
[14]McKenzie, R., McNulty, G. and Taylor, W., Algebras, lattices, varieties (Wadsworth Inc. and Brooks/Cole Publishing Co., 1987).Google Scholar
[15]Pixley, A.F., ‘The ternary discriminator function in universal algebra’, Math. Ann. 191 (1971), 167180.CrossRefGoogle Scholar
[16]Quackenbush, R., ‘Structure theory for equational classes generated by quasiprimal algebras’, Trans. Amer. Math. Soc. 187 (1974), 127145.CrossRefGoogle Scholar
[17]Taylor, W., ‘Residually small varieties’, Algebra Universalis 2 (1972), 3353.CrossRefGoogle Scholar