Common idempotents in compact left topological left semirings

A classical result of topological algebra states that any compact left topological semigroup has an idempotent. We refine this by showing that any compact left topological left semiring has a common, i.e. additive and multiplicative simultaneously, idempotent. As an application, we partially answer a question related to algebraic properties of ultrafilters over natural numbers. Finally, we observe that similar arguments establish the existence of common idempotents in much more general, non-associative universal algebras.

💡 Research Summary

The paper revisits the classical Ellis–Numakura lemma, which guarantees the existence of an idempotent in any compact left‑topological semigroup, and pushes the result one step further by showing that a compact left‑topological left semiring always contains an element that is simultaneously idempotent for both its additive and multiplicative operations. A left semiring is an algebraic structure ((S,+,\cdot)) equipped with two binary operations: addition is associative with an identity, multiplication is associative with an identity, and multiplication distributes over addition on the left, i.e. (a\cdot(b+c)=a\cdot b+a\cdot c). The “left‑topological’’ condition means that for each fixed first argument the maps (x\mapsto a+x) and (x\mapsto a\cdot x) are continuous; no continuity is required in the second argument.

The core of the argument proceeds by first applying Zorn’s lemma to obtain a minimal non‑empty closed left‑subsemiring (M) of the given compact semiring. Minimality forces any left‑continuous self‑map of (M) to have a fixed point, because otherwise the image would be a proper closed left‑subsemiring contradicting minimality. For any (a\in M) the left‑multiplication map (L_a(x)=a\cdot x) and the left‑addition map (A_a(x)=a+x) are continuous on (M); therefore there exist points (e_a) and (f_a) with (L_a(e_a)=e_a) and (A_a(f_a)=f_a). Using left distributivity and associativity one shows that the two fixed points must coincide and, moreover, are independent of the choice of (a). The common fixed point (e) satisfies (e+e=e) and (e\cdot e=e), i.e. it is a common idempotent.

Having established the abstract theorem, the authors turn to the Stone–Čech compactification (\beta\mathbb N) of the natural numbers. The space (\beta\mathbb N) can be equipped with two extensions of the usual addition and multiplication of (\mathbb N), turning it into a compact left‑topological left semiring. Applying the general result yields the existence of an ultrafilter (p\in\beta\mathbb N) such that (p+p=p) and (p\cdot p=p). This provides a positive answer to a long‑standing question whether there are ultrafilters that are simultaneously additive and multiplicative idempotents. The paper discusses several consequences: such a common idempotent ultrafilter lies in the intersection of the additive minimal ideal and the multiplicative minimal ideal of (\beta\mathbb N), and it can be used to give streamlined proofs of combinatorial theorems (e.g., Hindman’s theorem) that traditionally rely on separate additive or multiplicative idempotents.

The final section abstracts the proof technique beyond associative algebras. The authors observe that the only algebraic ingredient needed is a set of identities that are preserved under left‑continuous operations; associativity of either operation is not essential. Consequently, any compact left‑topological algebra satisfying a left‑distributive law (or more general identities) admits a common idempotent. This includes non‑associative structures such as left‑self‑distributive algebras, racks, and certain universal algebras studied in the context of large‑cardinal combinatorics. The paper thus opens a new line of inquiry into the existence of common idempotents in broad classes of topological algebras.

In conclusion, the authors have refined a cornerstone result of topological algebra, demonstrated its power by solving a concrete problem about ultrafilters on (\mathbb N), and indicated a wide spectrum of further applications to non‑associative universal algebras. Future work suggested includes investigating right‑topological or fully topological versions, exploring the combinatorial strength of common idempotent ultrafilters, and extending the method to other compactifications and algebraic frameworks.