Volume 61 Issue 2 (2011) - Годишник на ТУ - София - Технически ...

Proposition 3.8 For any n ≥ 2 in the semiring of idempotent elementsID (E An ) there is a chain of ideals ID 0 ⊂ ID 1 ⊂ · · · ⊂ ID n−2 ⊂ ID n−1 .Note that ID n−2 = ID (E An ) \{i} is a maximal ideal of this semiring.4 ConclusionIn the present paper the endomorphism semiring of a finite semilattice of aspecial type is investigated. The main results are Theorem 3.1 which gives thestructure of the center of the semiring of nonconstant endomorphisms and Theorem3.4 where we prove that idempotent elements formed a commutative subsemiringof the endomorphism semiring.References[1] Anderson A., Belnap N. (1975), Entailment, the Logic of Relevanceand Necessity, vol. I, Princeton Univ. Press, Princeton, 1975.[2] El Bashir R., Hurt J., Jan˘ca˘rík A., Kepka T. (2001), Simple commutativesemirings, J. Algebra 236 (2001), 277 – 306.[3] Golan J. (1999), Semirings and Their Applications, Kluwer, Dordrecht,1999.[4] Gratzer G. (2011), Lattice Theory: Foundation, Birkhäuser SpringerBasel AG, 2011.[5] Jeẑek J., Kepka T., Maròti M. (2009), The endomorphism semiring ofa semilattice, Semigroup Forum, 78 (2009), 21 – 26.[6] Kala V., Kepka T. (2008), A note on finitely generated ideal-simplecommutative semirings, Comment.Math.Univ.Carolin., 49, 1 (2008), 1 – 9.[7] Monico C. (2004), On finite congruence-simple semirings, J. Algebra271 (2004) 846 – 854.[8] Moore E. H. (1902), A definition of abstract groups, Trans. Amer.Math. Soc, 3 (1902), 485 – 492.[9] Trendafilov I., Vladeva D. (2011), The endomorphism semiring of afinite chain, Proc. Techn. Univ.-Sofia, 61 (2011).[10] Trendafilov I., Vladeva D. (2011), Endomorphism semirings withoutzero of a finite chain, Proc. Techn. Univ.-Sofia, 61 (2011).[11] Zumbrägel J. (2008), Classification of finite congruence-simplesemirings with zero, J. Algebra Appl. 7 (2008) 363 – 377.Authors: Ivan Trendafilov, assoc. prof., Department "Algebra andgeometry", FAMI, TU–Sofia, e-mail: ivan_d_trendafilov@abv.bgDimitrinka Vladeva, assoc. prof., Department "Mathematicsand physics", LTU, Sofia, e-mail: d_vladeva@abv.bgПостъпила на 05.12.2011Рецензент доц. д-р Георги Бижев28