Idempotents in Semigroups

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This is a presentation about the idempotent elements in semigroups and their relations with Ramsey Theory.

Text of Idempotents in Semigroups

  • Idempotents in compact semigroups and Ramsey Theory

  • Complete disorder is impossibleT.S.Motzkin

  • Piegenhole principleIf m objects are some colored with n colors and m>n then two objects have the same color.

  • Schurs theorem (1916)

  • Van der Waerdens Theorem (1927)

  • Ramseys Theorem (1930,finite version)

    Let r,k,l be given integers. Then there is a positive integer n with the following property.If the k-subsets of an n-set are colored with r colors,then there is a monochromatic l-set i.e one all of whose k-sets have the same color.

  • Ramseys theorem (infinite form)Let X be an infinite set, and k and r positive integers. Suppose that the k-subsets of X are colored with r colors. Then there is an infinite subset Y of X, all of whose k-subsets have the same color.

  • Definition .

  • Folkmans TheoremIf N is finitely colored there exist arbitrarily large finite sets A such that FS(A) is monochromatic.

  • Hindmans TheoremIf N is finitely colored there exists infinite such that FS(S) is monochromatic.

  • Hales-Jewett Theorem

  • Theorem of Milliken and Taylor

  • From now on Compact semigroup=compact hausdorf right topological semigroup

  • Theorem.Any compact semigroup has idempotent elements.

  • TheoremIf S is a compact semigroup then S has minimal left ideals.

  • Theorem

  • Theorem

  • Theorem

  • Finite Coloring

  • Lemma

  • Corollary

  • Theorem

  • Definition

  • Theorem

  • Corollaries :

  • Theorem

  • Variable Word

  • Theorem of Carlson and Simpson

  • Theorem

  • Corollary:

    I am very interested in Ramsey Theory.