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

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a connected algebraic monoid with group of units and lattice of regular -classes . The connection between the solvability of and the semilattice decomposition of into archimedean semigroups is further elaborated. If has a zero and if , then it is shown that is solvable if and only if is relatively complemented. If , then we introduce two basic numbers and and study their properties. Crucial to this process is the theorem that for any indempotent of , the centralizer of in 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 with central idempotents need not be conjugate. However special maximal connected submonoids with central idempotents are conjugate. If is regular, then is a Clifford semigroup if and only if for all , the set is finite. Finally the maximal semilattice image of any connected monoid is determined.

**[1]**Armand Borel,*Linear algebraic groups*, Notes taken by Hyman Bass, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR**0251042****[2]**Branko Grünbaum,*Convex polytopes*, With the cooperation of Victor Klee, M. A. Perles and G. C. Shephard. Pure and Applied Mathematics, Vol. 16, Interscience Publishers John Wiley & Sons, Inc., New York, 1967. MR**0226496****[3]**James E. Humphreys,*Linear algebraic groups*, Springer-Verlag, New York-Heidelberg, 1975. Graduate Texts in Mathematics, No. 21. MR**0396773****[4]**Mohan S. Putcha,*Semilattice decompositions of semigroups*, Semigroup Forum**6**(1973), no. 1, 12–34. MR**0369582**, https://doi.org/10.1007/BF02389104**[5]**Mohan S. Putcha,*On linear algebraic semigroups. I, II*, Trans. Amer. Math. Soc.**259**(1980), no. 2, 457–469, 471–491. MR**567091**, https://doi.org/10.1090/S0002-9947-1980-0567091-0**[6]**-,*On linear algebraic semigroups*. II, Trans. Amer. Math. Soc.**259**(1980), 471-491.**[7]**Mohan S. Putcha,*On linear algebraic semigroups. III*, Internat. J. Math. Math. Sci.**4**(1981), no. 4, 667–690. MR**663652**, https://doi.org/10.1155/S0161171281000513**[8]**-,*Green's relations on a connected algebraic monoid*, Linear and Multilinear Algebra (to appear).**[9]**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**, https://doi.org/10.1080/03081088208817469**[10]**Takayuki Tamura,*The theory of construction of finite semigroups. I*, Osaka Math. J.**8**(1956), 243–261. MR**0083497****[11]**Takayuki Tamura,*Another proof of a theorem concerning the greatest semilattice-decomposition of a semigroup*, Proc. Japan Acad.**40**(1964), 777–780. MR**0179282****[12]**Takayuki Tamura,*Note on the greatest semilattice decomposition of semigroups*, Semigroup Forum**4**(1972), 255–261. MR**0307990**, https://doi.org/10.1007/BF02570795

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1982-0662061-8

Keywords:
Connected monoids,
solvable groups,
-class,
lattice,
semilattice,
idempotents

Article copyright:
© Copyright 1982
American Mathematical Society