ON STABILITY IN SEPARATIVE SEMIGROUP
Abstract
Stability, as introduced by Koch and Wallace (1956), has long stood as a central
notion in semigroup theory, ensuring that inclusions of principal ideals collapse into
equalities and that left and right structures align harmoniously. Separativity, on
the other hand, generalises cancellativity while retaining algebraic regularity, and
Burmistrovich's decomposition theorem revealed that every separative semigroup
can be expressed as a semilattice of cancellative semigroups. Yet, whether separative semigroups inherit stability in the sense of Koch and Wallace has remained
unresolved. In this work, we close this gap: we prove that semilattices of cancellative semigroups are stable, and hence every separative semigroup is inherently
stable. This result elevates stability from a supplementary condition to a built-in
feature of separative semigroups, oering a unied perspective that strengthens the
foundations of semigroup theory and deepens its structural coherence.
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