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Representation of Semilattice of Right Zero Semigroups by Transformation Semigroup. P. 107–114

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Section: Physics. Mathematics. Informatics

UDC

512.533

Authors

Zyablitseva Larisa Vladimirovna
Institute of Mathematics, Information and Space Technologies, Northern (Arctic) Federal University named after M.V. Lomonosov (Arkhangelsk, Russia)
e-mail: zlarisav@yandex.ru

Abstract

Studying a semigroup to determine its structure, it would be convenient to consider the accurate representation of this semigroup by transformation semigroup. In this case, semigroup elements can be presented in the form of corresponding transformations. In some cases, when the semigroup is rather sophisticated, this representation can be the only easy way to describe it. The paper considers the idempotent semigroup S, which is a semilattice n of right zero semigroups. Earlier the author had studied the exact matrix representations of this semigroup and the corresponding transformation semigroups, though not of the entire semigroup S but only the transformations of the subsemigroup, which is the minimal ideal of this semigroup. The paper proves that homomorphism F: S → ℑ(S), acting by the rule: F(u) = ru ( ru – right shift corresponding to the element u ∈ S) is a faithful representation of the semigroup S by transformation semigroup. In addition, the author has studied the type of elements obtained at this mapping.

Keywords

idempotent semigroup, semilattice, representation of semigroups, transformation semigroup
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References

  1. Artamonov V.A., Saliy V.N., Skornyakov L.A. Obshchaya algebra [General Algebra]. Moscow, 1991.
  2. Zyablitseva L.V. Tochnye matrichnye predstavleniya svyazok, yavlyayushchikhsya polureshetkami polugrupp pravykh nuley [Accurate Matrix Representations of Bands Being Semilattices of Right Zero Semigroups]. Sovremennaya algebra [Modern Algebra]. Iss. 3 (33). Rostov-on-Don, 1998, pp. 48–55.