Factorizations of infinite sequences in semigroups (Q1568758)

Let \(S\) be a semigroup and \(s=s_1,s_2,\dots\) a finite or infinite sequence of elements of \(S\). A finite subsequence of consecutive elements \(s_i,s_{i+1},\dots,s_{i+k}\) is called a segment of \(s\) with value \(s_is_{i+1}\cdots s_{i+k}\). An \(m\)-factorization of \(s\) is a sequence \(t_1,\dots,t_m\) where the \(t_j\) are values of consecutive segments of \(s\). Note that an \(m\)-factorization may let aside some left part and some right part of \(s\). Similarly, when \(s\) is infinite, an \(\omega\)-factorization of \(s\) is an infinite sequence \(t_1,t_2,\dots\) where the \(t_j\) are values of consecutive segments of \(s\). A well-known corollary to the Ramsey Theorem tells us that in every finite semigroup each infinite sequence has an \(\omega\)-factorization of the form \(e,e,\dots\) where \(e\) is an idempotent of \(S\). Accordingly, an infinite sequence is called Ramseyan if it admits an \(\omega\)-factorization of this form, and a semigroup \(S\) is called infinitely Ramseyan if all infinite sequences formed of its elements are Ramseyan. Hence every finite semigroup is infinitely Ramseyan, and this property is useful in proving a number of combinatorial results on semigroups. The main result of the present paper gives a structural characterization of infinitely Ramseyan semigroups: a semigroup \(S\) is infinitely Ramseyan if and only if \(S\) is periodic and satisfies the descending chain condition for principal right ideals, all chains of idempotents of \(S\) are finite, all nil Rees factors of \(S\) contain nonzero left annihilators, and all completely 0-simple Rees factors of \(S\) are finite. In a similar way, one can define two-sided infinitely Ramseyan semigroups by considering two-sided infinite sequences \(\dots,s_{-2},s_{-1},s_1,s_2,\dots\), and then it is proved that a semigroup is two-sided infinitely Ramseyan if and only if it is left and right \(T\)-nilpotent, where left [right] \(T\)-nilpotence means that every right [left] infinite sequence has a segment with value zero.