Connected algebraic monoids
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- by Mohan S. Putcha PDF
- Trans. Amer. Math. Soc. 272 (1982), 693-709 Request permission
Abstract:
Let $S$ be a connected algebraic monoid with group of units $G$ and lattice of regular $\mathcal {J}$-classes $\mathcal {U}(S)$. The connection between the solvability of $G$ and the semilattice decomposition of $S$ into archimedean semigroups is further elaborated. If $S$ has a zero and if $\mathcal {U}(S)\le 7$, then it is shown that $G$ is solvable if and only if $\mathcal {U}(S)$ is relatively complemented. If $J\in \mathcal {U}(S)$, then we introduce two basic numbers $\theta (J)$ and $\delta (J)$ and study their properties. Crucial to this process is the theorem that for any indempotent $e$ of $S$, the centralizer of $e$ in $G$ is connected. Connected monoids with central idempotents are also studied. A conjecture about their structure is forwarded. It is pointed out that the maximal connected submonoids of $S$ with central idempotents need not be conjugate. However special maximal connected submonoids with central idempotents are conjugate. If $S$ is regular, then $S$ is a Clifford semigroup if and only if for all $f\in E(S)$, the set $\{ e|e \in E(S), e \geq f\}$ is finite. Finally the maximal semilattice image of any connected monoid is determined.References
- Armand Borel, Linear algebraic groups, W. A. Benjamin, Inc., New York-Amsterdam, 1969. Notes taken by Hyman Bass. MR 0251042
- Branko Grünbaum, Convex polytopes, Pure and Applied Mathematics, Vol. 16, Interscience Publishers John Wiley & Sons, Inc., New York, 1967. With the cooperation of Victor Klee, M. A. Perles and G. C. Shephard. MR 0226496
- James E. Humphreys, Linear algebraic groups, Graduate Texts in Mathematics, No. 21, Springer-Verlag, New York-Heidelberg, 1975. MR 0396773
- Mohan S. Putcha, Semilattice decompositions of semigroups, Semigroup Forum 6 (1973), no. 1, 12–34. MR 369582, DOI 10.1007/BF02389104
- Mohan S. Putcha, On linear algebraic semigroups. I, II, Trans. Amer. Math. Soc. 259 (1980), no. 2, 457–469, 471–491. MR 567091, DOI 10.1090/S0002-9947-1980-0567091-0 —, On linear algebraic semigroups. II, Trans. Amer. Math. Soc. 259 (1980), 471-491.
- Mohan S. Putcha, On linear algebraic semigroups. III, Internat. J. Math. Math. Sci. 4 (1981), no. 4, 667–690. MR 663652, DOI 10.1155/S0161171281000513 —, Green’s relations on a connected algebraic monoid, Linear and Multilinear Algebra (to appear).
- Mohan S. Putcha, The group of units of a connected algebraic monoid, Linear and Multilinear Algebra 12 (1982/83), no. 1, 37–50. MR 672915, DOI 10.1080/03081088208817469
- Takayuki Tamura, The theory of construction of finite semigroups. I, Osaka Math. J. 8 (1956), 243–261. MR 83497
- Takayuki Tamura, Another proof of a theorem concerning the greatest semilattice-decomposition of a semigroup, Proc. Japan Acad. 40 (1964), 777–780. MR 179282
- Takayuki Tamura, Note on the greatest semilattice decomposition of semigroups, Semigroup Forum 4 (1972), 255–261. MR 307990, DOI 10.1007/BF02570795
Additional Information
- © Copyright 1982 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 272 (1982), 693-709
- MSC: Primary 20M10; Secondary 20G99
- DOI: https://doi.org/10.1090/S0002-9947-1982-0662061-8
- MathSciNet review: 662061