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Every nonempty compact Hausdorff space carrying a left-topological semigroup structure contains an idempotent element of a semigroup.

Choose an inclusion-minimal nonempty compact subsemigroup $K$, using Zorn's lemma and the finite intersection property. For $a\in K$, the set $aK$ is a nonempty compact subsemigroup, so minimality gives $aK=K$. Hence $F=\{x\in K:ax=a\}$ is nonempty. It is a compact closed subset of $K$ by continuity and the Hausdorff space assumption, and is a subsemigroup because $a(xy)=(ax)y=ay=a$ for $x,y\in F$. Minimality gives $F=K$, so $a^2=a$.^[ref]

Reversing multiplication gives the same conclusion for a right-topological semigroup. Joint continuity of multiplication is not needed.

A left-topological semigroup is a semigroup with a topology for which every left translation $L_a:x\mapsto ax$ is a continuous map. Here $a$ is fixed and the right input varies. Requiring continuous right translations $R_a:x\mapsto xa$ instead gives a right-topological semigroup.^[ref]

Vitaly Bergelson studies the interaction of ergodic theory with Ramsey theory and arithmetic patterns. His Ohio State publication list documents work on recurrence, polynomial configurations and semigroup actions.^[ref]

His Ergodic Ramsey Theory - an update by Vitaly Bergelson develops the exchange between recurrence in dynamical systems and structured configurations in subsets of integers. With Neil Hindman and Kendall Williams, he also extended Milliken–Taylor theorem patterns to polynomial settings, using algebraic properties of ultrafilters to obtain monochromatic configurations.^[ref][ref]