The Bergman property for semigroups

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V. MALTCEV, J. D. MITCHELL, AND N. RUŠKUC
st ∈ U i and U i is a subsemigroup. If x = y = 1 S and t ∈ U i , then xty ∈ T and so t = xty ∈ T i .
Hence U i is contained in T i and as such is a proper subsemigroup of T .
Now, let us construct a cofinal chain for S using the chain (U i ) i ℵ 0 . Then if scf(T ) > ℵ 0 , we could deduce that scf(T ) =
cf(T ) = cf(S) = scf(S), by Proposition 2.2(ii) and the first part of the theorem. So, we are left
with the task of proving that scf(T ) > ℵ 0 .
Let Ψ : T → N be any function satisfying
(st)Ψ ≤ (s)Ψ + (t)Ψ + k Ψ
for all s, t ∈ T and for some constant k Ψ ∈ {0, 1, 2, . . .}. By Lemma 2.3, it suffices to prove
that Ψ is bounded. We proceed in a similar fashion as in the proof of the previous part of the
theorem. That is, we define Φ : T → N using Ψ and subsequently define Υ : S → N satisfying
(2.1). Let Φ : T → N be defined by
(t)Φ = max{ (xty)Ψ : x, y ∈ S \ T, xty ∈ T }.
Note that Φ is well-defined since the set { (xty)Ψ : x, y ∈ S \ T, xty ∈ T } is non-empty and
finite. To prove that Φ satisfies (2.1) let s, t ∈ T . Then
(st)Φ = max{ (x · st · y)Ψ : x, y ∈ S \ T, x · st · y ∈ T }.
The set { (x · st · y)Ψ : x, y ∈ S \ T, x · st · y ∈ T } is the union of the following three sets
So,
and Φ satisfies (2.1).
A = { (xs · t · y)Ψ : x, y ∈ S \ T, xs · t · y ∈ T, xs ∈ S \ T },
B = { (x · s · ty)Ψ : x, y ∈ S \ T, x · s · ty ∈ T, ty ∈ S \ T },
C = { (xs · ty)Ψ : x, y ∈ S \ T, xs · ty ∈ T, xs, ty ∈ T }.
(st)Φ ≤ max{ max A, max B, max C }
≤ max{(t)Φ, (s)Φ, (s)Φ + (t)Φ + k Ψ } = (s)Φ + (t)Φ + k Ψ ,