# 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.

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