On linear algebraic semigroups. II

Author:
Mohan S. Putcha

Journal:
Trans. Amer. Math. Soc. **259** (1980), 471-491

MSC:
Primary 20M10

DOI:
https://doi.org/10.1090/S0002-9947-80-99945-6

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Abstract: We continue from [**11**] the study of linear algebraic semigroups. Let *S* be a connected algebraic semigroup defined over an algebraically closed field *K*. Let be the partially ordered set of regular -classes of *S* and let be the set of idempotents of *S*. The following theorems (among others) are proved. (1) *is a finite lattice*. (2) *If S is regular and the kernel of S is a group, then the maximal semilattice image of S is isomorphic to the center of* . (3) *If S is a Clifford semigroup and* , *then the set* *is finite*. (4) *If S is a Clifford semigroup, then there is a commutative connected closed Clifford subsemigroup T of S with zero such that T intersects each* -*class of S*. (5) *If S is a Clifford semigroup with zero, then S is commutative and is in fact embeddable in* *for some* . (6) *If* *and S is a commutative Clifford semigroup, then S is isomorphic to a direct product of an abelian connected unipotent group and a closed connected subsemigroup of* *for some* . (7) *If S is a regular semigroup and* , *then* . (8) *If S is a Clifford semigroup with zero and* , *then* *can be any even number* . (9) *If S is a Clifford semigroup then* *is a relatively complemented lattice and all maximal chains in* *have the same number of elements*.

**[1]**Armand Borel,*Linear algebraic groups*, Notes taken by Hyman Bass, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR**0251042****[2]**A. H. Clifford,*Semigroups admitting relative inverses*, Ann. of Math. (2)**42**(1941), 1037–1049. MR**0005744**, https://doi.org/10.2307/1968781**[3]**A. H. Clifford and G. B. Preston,*The algebraic theory of semigroups. Vol. I*, Mathematical Surveys, No. 7, American Mathematical Society, Providence, R.I., 1961. MR**0132791****[4]**P. Crawley and R. P. Dilworth,*Algebraic theory of lattices*, Prentice-Hall, Englewood Cliffs, N. J., 1973.**[5]**Michel Demazure and Pierre Gabriel,*Groupes algébriques. Tome I: Géométrie algébrique, généralités, groupes commutatifs*, Masson & Cie, Éditeur, Paris; North-Holland Publishing Co., Amsterdam, 1970 (French). Avec un appendice Corps de classes local par Michiel Hazewinkel. MR**0302656****[6]**M. P. Drazin,*Natural structures on rings and semigroups with involution*(to appear).**[7]**T. E. Hall,*The partially ordered set of all 𝐽-classes of a finite semigroup*, Semigroup Forum**6**(1973), no. 3, 263–264. MR**0393301**, https://doi.org/10.1007/BF02389131**[8]**Kenneth Hoffman and Ray Kunze,*Linear algebra*, Second edition, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1971. MR**0276251****[9]**James E. Humphreys,*Linear algebraic groups*, Springer-Verlag, New York-Heidelberg, 1975. Graduate Texts in Mathematics, No. 21. MR**0396773****[10]**W. D. Munn,*Pseudo-inverses in semigroups*, Proc. Cambridge Philos. Soc.**57**(1961), 247–250. MR**0121410****[11]**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**[12]**J. Rhodes,*Problems*23-28, Semigroup Forum**5**(1972), 92-94.**[13]**I. R. Shafarevich,*Basic algebraic geometry*, Springer-Verlag, New York-Heidelberg, 1974. Translated from the Russian by K. A. Hirsch; Die Grundlehren der mathematischen Wissenschaften, Band 213. MR**0366917****[14]**Takayuki Tamura,*The theory of construction of finite semigroups. I*, Osaka Math. J.**8**(1956), 243–261. MR**0083497**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-80-99945-6

Keywords:
Linear algebraic semigroup,
idempotent,
subgroup,
-class

Article copyright:
© Copyright 1980
American Mathematical Society