CLIFFORD SEMIGROUPS AND SEMINEAR-RINGS OF ENDOMORPHISMS

Year 2010, Volume: 7 Issue: 7, 110 - 119, 01.06.2010

Nick D. Gilbert Mohammad Samman

Abstract

We consider the structure of the semigroup of self-mappings of a semigroup S under pointwise composition, generated by the endomorphisms of S. We show that if S is a Clifford semigroup, with underlying semilattice Λ, then the endomorphisms of S generate a Clifford semigroup E+(S) whose underlying semilattice is the set of endomorphisms of Λ. These results contribute to the wider theory of seminear-rings of endomorphisms, since E+(S) has a natural structure as a distributively generated seminear-ring.

Keywords

endomorphism, Clifford semigroup, seminear-ring

References

• J. C. M. Baeten and W. P. Weijland, Process Algebra, Cambridge Tracts in Theoretical Computer Science 18, Cambridge University Press, 1990. T. Boykett, Seminearring of computation, preprint, citeseer.ist.psu.edu/368766.html ) . Linz, ( Published electronically at
• A. Fr¨ohlich, The near-ring generated by the inner automorphisms of a finite simple group, J. London Math. Soc., 33 (1958), 95-107.
• N. D. Gilbert and M. Samman, Endomorphism seminear-rings of Brandt semi- groups, Comm. Algebra (to appear). J. S. Golan, Semirings and Their Applications, Kluwer Academic Press, 1999.
• J. Je˘zek, T. Kepka and M. Mar´oti, The endomorphism ring of a semilattice, Semigroup Forum, 78 (2009), 21-26.
• C. G. Lyons and J. J. Malone, Endomorphism near-rings, Proc. Edin. Math. Soc., 17 (1970), 71-78.
• J. D. P. Meldrum, The endomorphism near-rings of finite general linear groups, Proc. Royal Irish Acad., 79A (1979), 87-96.
• J. D. P. Meldrum, Near-rings and Their Links with Groups, Research Notes in Math. 134, Pitman Publishing Ltd., 1985.
• J. D. P. Meldrum and M. Samman, On free d.g. seminear-rings, Riv. Mat. Univ. Parma (5), 6 (1997), 93-102.
• J. D. P. Meldrum and M. Samman, On endomorphisms of semilattices of groups, Algebra Colloq., 12 (2005), 93-100.
• H. Neumann, On varieties of groups and their associated near-rings, Math. Z., (1956), 36-69.
• M. Samman, Topics in Seminear-ring Theory, PhD Thesis, University of Ed- inburgh, 1998. Nick D. Gilbert
• School of Mathematical and Computer Sciences & the Maxwell Institute for the Mathematical Sciences, Heriot-Watt University, Edinburgh EH14 4AS, U.K. e-mail: [e-mail] Mohammad Samman Department of Mathematics and Statistics King Fahd University of Petroleum & Minerals Box 411, Dhahran 31261, Saudi Arabia e-mail: [e-mail]