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Maps into Dynkin diagrams arising from regular monoids

Part of: Linear algebraic groups and related topics Semigroups

Published online by Cambridge University Press:  09 April 2009

M. K. Augustine  and
Mohan S. Putcha
M. K. Augustine
Affiliation:
Department of Mathematics, North Carolina State University Raleigh, North Carolina 27695-8205, U.S.A.
Mohan S. Putcha
Affiliation:
Department of Mathematics, North Carolina State University Raleigh, North Carolina 27695-8205, U.S.A.
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Abstract

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It has been shown by one of the authors that the system of idempotents of monoids on a group G of Lie type with Dynkin diagram Γ can be classified by the following data: a partially ordered set U with maximum element 1 and a map λ: U → 2Γ with λ(1) = Γ and with the property that for all J1, J2, J3 ∈ U with J1 > J2 > J3, any connected component of λ(J2) is contained in either λ(J1) or λ(J3). In this paper we show that λ comes from a regular monoid if and only if the following conditions are satisfied: (1) U is a ∧-semilattice; (2) If J1, J2 ∈ U, then λ(J1)∧ λ(J2) λ(J1 ∧ J2); (3) If θ ∈ Γ, J ∈ U, then max{J1 ∈ U|J1 > J, θ ∈ λ (J1)} exists; (4) If J1, J2 ∈ U with J1 > J2 and if X is a two element discrete subset of λ(J1) ∪ λ(J2), then X λ(J) for some J ∈ UJ with J1 > J > J2.

MSC classification

Secondary: 20G99: None of the above, but in this section 20M17: Regular semigroups
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 47 , Issue 2 , October 1989 , pp. 313 - 321
Copyright
Copyright © Australian Mathematical Society 1989

References

[1]Misra, K. C., Putcha, M. S. and Singh, D. S., ‘Transitive maps from posets to Dynkin diagrams’, to appear.Google Scholar
[2]Nambooripad, K. S. S., ‘Structure of regular semigroups I.’, Mem. Amer. Math. Soc. 224 (1979).Google Scholar
[3]Putcha, M. S., ‘A semigroup approach to linear algebraic groups’, J. Algebra 80 (1983), 164–185.CrossRefGoogle Scholar
[4]Putcha, M. S., Linear Algebraic Monoids, (London Math. Soc. Lecture Note Series, No. 133, Cambridge Univ. Press, 1988).CrossRefGoogle Scholar
[5]Putcha, M. S., ‘Monoids on groups with BN-pairs’, J. Algebra 120 (1989), 139–169.CrossRefGoogle Scholar
[6]Putcha, M. S. and Renner, L. E., ‘The system of idempotents and the lattices of-classes of reductive algebraic monoids’, J. Algebra 116 (1988), 385–399.CrossRefGoogle Scholar
[7]Renner, L. E., ‘Analogue of the Bruhat decomposition for algebraic monoids’, J. Algebra 101 (1986), 303–338.CrossRefGoogle Scholar
[8]Suzuki, M., Group Theory I, (Springer-Verlag, 1982).CrossRefGoogle Scholar