The Arithmetic of the Quasi-Uniserial Semigroups without Zero

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Abstract
An element a in a partially ordered semigroup T is called integral if is valid. The integral elements form a subsemigroup S of T if they exist. Two different integral idempotents e and f in T generate different one-sided ideals, because eT = fT, say, implies e = fe ⊆ f and f = ef ⊆ e.Let M be a completely simple semigroup. M is the disjoint union of its maximal subgroups [4]. Their identity elements generate the minimal one-sided ideals in M. The previous paragraph suggests the introduction of the following hypothesis on M.Hypothesis 1. Every minimal one-sided ideal in M is generated by an integral idempotent.

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