[1] Shi Xiaoping,. On S-acts containing no maximal S-subacts [J]. Journal of Southeast University (English Edition), 2006, 22 (4): 593-596. [doi:10.3969/j.issn.1003-7985.2006.04.030]

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On S-acts containing no maximal S-subacts()

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Journal of Southeast University (English Edition)[ISSN:1003-7985/CN:32-1325/N]

Volumn:

22

Issue:

2006 4

Page:

593-596

Research Field:

Mathematics, Physics, Mechanics

Publishing date:

2006-12-30

Info

Title:

On S-acts containing no maximal S-subacts

Author(s):

Shi Xiaoping

School of Sciences, Southeast University, Nanjing 210096, China

Keywords:

S-act;  maximal subact;  C-subact

PACS:

O152.7

DOI:

10.3969/j.issn.1003-7985.2006.04.030

Abstract:

Let S be a semigroup and let A be an S-act.Some necessary and sufficient conditions that S-subacts of A are maximal S-subacts are given.A relation L which is similar to the Green relation in semigroups is defined.By the relation L, it is proved that a non-empty set L of A is a maximal S-subact if and only if A\L is a(maximal)L-class.Finally, the concept of a C-subact is defined, some properties of C-subacts are discussed, and it is proved that A contains no maximal S-subacts if and only if every cyclic S-subact of A is a C-subact.Consequently, the results obtained by Imrich Fabrici that semigroups contain no maximal(left)ideals are the corollary of this paper.

References:

[1] Kilp M, Knauer U, Mikhalev A V.Monoids, acts and categories, with applications to wreath products and graphs[M].New York:Walter de Gruyter, 2000.
[2] Liu Z K.Theory of S-acts over semigroups[M].Beijing:Science Press, 1998.(in Chinese)
[3] Fabrici I.Semigroups containing covered one-sided ideals[J].Math Slovaca, 1981, 31:225-231.
[4] Fabrici I.Semigroups containing no maximal ideals[J].Semigroup Forum, 1990, 40:101-104.
[5] Howie J M.Fundamentals of semigroup theory [M].Oxford:Oxford University Press, 1995.

Memo

Memo:

Biography: Shi Xiaoping(1975—), male, doctor, sxp72@sohu.com.

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