×

Ordered semigroups having the \(P\)-property. (English. Russian original) Zbl 1178.06010

A semigroup \(S\) is an Archimedean semigroup if \(\forall a,b\in S\), \(b^m\in I(a)\) and \(a^n\in I(b)\) for some \(m,n\in N\), where \(I(a)\) is the principal ideal of \(S\) generated by \(a\). Chrislock, Putcha and Tamura established many results about such semigroups; e.g., Tamura proved that \(S\) is a semilattice of Archmidean groups if and only if it has the power property \((P\)-property). He also introduced the \(P_m\)-property.
In this paper, the authors generalize these works for ordered semigroups using more simplified techniques. They start by proving this simple fact for an ordered Archimedean semigroup \(S\): \(\forall a,b\in S,\exists k\in N\) such that \(b^k\in I(a)\). After defining the \(P\)-property for ordered semigroups and deducing that every Archimedean ordered semigroup has the \(P\)-property, they proceed to prove their first theorem: An ordered semigroup \(S\) has the \(P\)-property if and only if it is a complete semilattice of semigroups with the \(P\)-property. CS-indecomposable ordered semigroups are defined, and Theorem 2 proved: An ordered semigroup is CS-indecomposable and has the \(P\)-property if and only if it is Archimedean.
Finally the authors show in Theorem 3 that if the ordered semigroup \(S\) has the \(P_m\)-property for certain \(m\) \((\geq 2)\in N\) then \(S\) has the \(\lambda\)-property \(\forall\lambda\in N\).

MSC:

06F05 Ordered semigroups and monoids
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] A. H. Clifford and G. B. Preston, The Algebraic Theory of Semigroups, Amer. Math. Soc., Math. Surveys (Providence, Rhode Island, 1964), Vol. 7. · Zbl 0111.03403
[2] J. L. Chrislock, ”On Medial Semigroups,” J. Algebra 12, 1–9 (1969). · Zbl 0187.29102 · doi:10.1016/0021-8693(69)90013-1
[3] M. S. Putcha, ”Semilattice Decompositions of Semigroups,” Semigroup Forum 6(1), 12–34 (1973). · Zbl 0256.20074 · doi:10.1007/BF02389104
[4] T. Tamura ”On Putcha’s Theorem Concerning Semilattice of Archimedian Semigroups,” Semigroup Forum 4, 83–86 (1972). · Zbl 0256.20075 · doi:10.1007/BF02570773
[5] N. Kehayopulu, ”On Weakly Prime Ideals of Ordered Semigroups,” Math. Japonica 35(6), 1051–1056 (1990). · Zbl 0717.06006
[6] N. Kehayopulu, ”On weakly Commutative poe-Semigroups,” Semigroup Forum 34(3), 367–370 (1987). · Zbl 0613.06009 · doi:10.1007/BF02573174
[7] N. Kehayopulu, ”Remark on Ordered Semigroups,” Math. Japonica 35(6), 1061–1063 (1990). · Zbl 0717.06008
[8] N. Kehayopulu and M. Tsingelis, ”Remark on Ordered Semigroups,” in Decompositions and Homomorphic Mappings of Semigroups, Ed. by E. S. Lyapin (Interuniversitary collection of scientific works, St. Petersburg, Obrazovanie, 1992), pp. 50–55. · Zbl 0823.06006
[9] N. Kehayopulu and M. Tsingelis, ”A Note on Semilattice Congruences in Ordered Semigroups,” Izv. Vyssh. Uchebn. Zaved., Mat. No. 2, 50–52 [Russian Mathematics (Iz. VUZ) 44 (2), 48–50 (2000)]. · Zbl 0976.06009
[10] N. Kehayopulu, P. Kiriakuli, Rao S. Hanumantha, and P. Lakshmi, ”On Weakly Commutative poe-Semigroups,” Semigroup Forum 41(3), 272–276 (1990). · Zbl 0708.06011 · doi:10.1007/BF02573402
[11] N. Kehayopulu and M. Tsingelis, ”On Weakly Commutative Ordered Semigroups,” Semigroup Forum 56(1), 32–35 (1998). · Zbl 0890.06010 · doi:10.1007/s00233-002-7002-6
[12] N. Kehayopulu and M. Tsingelis, ”Semilattices of Archimedian Ordered Semigroups,” Algebra Colloquium 15(3), 527–540 (2008). · Zbl 1160.06005 · doi:10.1142/S1005386708000527
[13] N. Kehayopulu and M. Tsingelis, ”CS-Indecomposable Ordered Semigroups,” Zap. Nauchn. Semin. POMI 343, pp. 222–232 (2007).
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.