Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Connected algebraic monoids


Author: Mohan S. Putcha
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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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\vert e \in E(S),\,e \geq f\} $ is finite. Finally the maximal semilattice image of any connected monoid is determined.


References [Enhancements On Off] (What's this?)

  • [1] A. Borel, Linear algebraic groups, Benjamin, New York, 1969. MR 0251042 (40:4273)
  • [2] B. Grünbaum, Convex polytopes, Interscience-Wiley, New York, 1967. MR 0226496 (37:2085)
  • [3] J. E. Humphreys, Linear algebraic groups, Springer-Verlag, Berlin and New York, 1975. MR 0396773 (53:633)
  • [4] M. S. Putcha, Semilattice decompositions of semigroups, Semigroup Forum 6 (1973), 12-34. MR 0369582 (51:5815)
  • [5] -, On linear algebraic semigroups, Trans. Amer. Math. Soc. 259 (1980), 457-469. MR 567091 (81i:20087)
  • [6] -, On linear algebraic semigroups. II, Trans. Amer. Math. Soc. 259 (1980), 471-491.
  • [7] -, On linear algebraic subgroups. III, Internat. J. Math. Math. Sci. 4 (1981), 667-690; Corrigendum 5 (1982), 205-207. MR 663652 (83k:20073a)
  • [8] -, Green's relations on a connected algebraic monoid, Linear and Multilinear Algebra (to appear).
  • [9] -, The group of units of a connected algebraic monoid, Linear and Multilinear Algebra (to appear). MR 672915 (84d:20065)
  • [10] T. Tamura, The theory of construction of finite semigroups. I, Osaka J. Math. 8 (1956), 243-261. MR 0083497 (18:717e)
  • [11] -, Another proof of a theorem concerning the greatest semilattice decomposition of a semigroup, Proc. Japan Acad. Ser. A. Math. Sci. 40 (1964), 777-780. MR 0179282 (31:3530)
  • [12] -, Note on the greatest semilattice decomposition of semigroups, Semigroup Forum 4 (1972), 255-261. MR 0307990 (46:7105)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 20M10, 20G99

Retrieve articles in all journals with MSC: 20M10, 20G99


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1982-0662061-8
Keywords: Connected monoids, solvable groups, $ \mathcal{J}$-class, lattice, semilattice, idempotents
Article copyright: © Copyright 1982 American Mathematical Society

American Mathematical Society