×
Satko, Ladislav; Grošek, Otokar

On SLA-ideals. (English) Zbl 0840.20065

Math. Slovaca 45, No. 1, 45-52 (1995).
A semigroup left \(A\)-ideal (SLA-ideal) of a semigroup \(S\) is a subsemigroup \(X\) of \(S\) such that \(sX\cap X\neq\emptyset\) for every \(s\in S\). (SLA-ideals are the same as M. Putcha’s mild ideals [Proc. Am. Math. Soc. 47, 49-52 (1975; Zbl 0307.20036)].) A semigroup is said to be SLA-simple if it has no proper SLA-ideals. The main result of the paper reads as follows: A semigroup has a minimal SLA-ideal if and only if it has a kernel which is a rectangular band of SLA-simple groups; in the commutative case the latter condition reduces to the kernel being a periodic group.
Reviewer: L.Márki (Budapest)

Cited in 1 Document

MSC:

20M12 Ideal theory for semigroups

Keywords:

left \(A\)-ideals of semigroups; mild ideals; SLA-ideals; minimal SLA-ideals; rectangular band of SLA-simple groups

Citations:

Zbl 0307.20036
PDFBibTeX XMLCite
\textit{L. Satko} and \textit{O. Grošek}, Math. Slovaca 45, No. 1, 45--52 (1995; Zbl 0840.20065)
Full Text: EuDML

References:

[1] BRUCK R. H.: Difference sets in a finite group. Trans. Amer. Math. Soc. 78 (1955), 464-484. · Zbl 0065.13302 · doi:10.2307/1993074
[2] CLIFFORD A. H.: Partially ordered Abelian groups. Ann. of Math. 41 (1940), 465-473. · Zbl 0025.00801 · doi:10.2307/1968728
[3] CLIFFORD A. H., PRESTON G. B.: The Algebraic Theory of Semigroups I; II. Amer. Math. Soc., Providence, R. I., 1961; 1967. · Zbl 0111.03403
[4] FUCHS L.: Partially Ordered Algebraic Systems. Pergamon Press, Oxford, 1963. · Zbl 0137.02001
[5] GROŠEK O., JAJCAY R.: Generalized difference sets on an infinite cyclic semigroup. J. Combin. Math. Combin. Comput. 13 (1993), 167-174. · Zbl 0777.05025
[6] GROŠEK O., SATKO L.: A new notion in the theory of semigroups. Semigroup Forum 20 (1980), 233-240. · Zbl 0439.20045 · doi:10.1007/BF02572683
[7] GROŠEK O., SATKO L.: Smallest A-ideals in semigroups. Semigroup Forum 23 (1981), 297-309. · Zbl 0484.20030 · doi:10.1007/BF02676654
[8] LYAPIN E. S., AIZENSHTAT A. YA., LESOKHIN M. M.: Exercises in Group Theory. Plenum Press, New York, 1972.
[9] PUTCHA M. S.: Maximal cancellative subsemigroups and cancellative congruences. Proc. Amer. Math. Soc. 47 (1975), 49-52. · Zbl 0307.20036 · doi:10.2307/2040206
[10] RANKIN S. A., REIS, C M.: Semigroups with quasi-zeros. Canad. J. Math. XXXII (1980), 511-530. · Zbl 0439.20040 · doi:10.4153/CJM-1980-040-x
[11] SATKO L., GROŠEK O.: On minimal A-ideals of semigroups. Semigroup Forum 23 (1981), 283-295. · Zbl 0477.20047 · doi:10.1007/BF02676653
[12] SATKO L., GROŠEK O.: On maximal A-ideals in semigroups. Colloq. Math. Soc. János Bolyai 39 (1981), 389-395.
[13] SCHWARZ Š.: On the structure of simple semigroups without zero. Czechoslovak Math. J. 1(76) (1951), 41-53. · Zbl 0045.15608
[14] SHIMBIREVA H.: On the theory of partially ordered groups. Mat. Sb. 20 (1947), 145-178 · Zbl 0029.10301
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.