A note on the congruence lattice of a finitely generated algebra
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- by Ivan Rival and Bill Sands
- Proc. Amer. Math. Soc. 72 (1978), 451-455
- DOI: https://doi.org/10.1090/S0002-9939-1978-0509233-5
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Abstract:
Let $\mathfrak {A}$ be a finitely generated algebra of finite type. If $\theta$ is a congruence relation of $\mathfrak {A}$ such that $\mathfrak {A}/\theta$ is finite then $\theta$ is compact in the lattice ${\text {Con}}(\mathfrak {A})$ of all congruence relations of $\mathfrak {A}$. Moreover, if $\mathfrak {A}$ is infinite then there is a congruence relation $\theta$ such that $\mathfrak {A}/\theta$ is infinite and $\mathfrak {A}/\theta ’$ is finite for every $\theta ’ > \theta$ in ${\text {Con}}(\mathfrak {A})$.References
- George Grätzer, Universal algebra, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1968. MR 0248066
- G. Grätzer and E. T. Schmidt, Two notes on lattice-congruences, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 1 (1958), 83–87. MR 99943
- Bjarni Jónsson, Topics in universal algebra, Lecture Notes in Mathematics, Vol. 250, Springer-Verlag, Berlin-New York, 1972. MR 0345895
- Donald Monk, On pseudo-simple universal algebras, Proc. Amer. Math. Soc. 13 (1962), 543–546. MR 144840, DOI 10.1090/S0002-9939-1962-0144840-1
Bibliographic Information
- © Copyright 1978 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 72 (1978), 451-455
- MSC: Primary 08A30
- DOI: https://doi.org/10.1090/S0002-9939-1978-0509233-5
- MathSciNet review: 509233